Skin Friction Drag Reduction in Turbulent Flows Using Superhydrophobic Surfaces by Anoop Rajappan B. Tech., Indian Institute of Technology Madras (2015)

Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering MAMU E OF TEHNO OG)Y at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY S 22 201l LIBRARIES September 2017 'ARCHNVES

@ 2017 Massachusetts Institute of Technology. All rights reserved. Signature redacted Signature of Author ...... Department 1f M'fechanical Engineering August 11, 2017 Signature redacted C ertified by ...... Gareth H4 I0kinley{ SoE Professor of Teaching 'Tnovation Thesis Supervisor Signature redacted A ccepted by ...... Rohan Abeyaratne Quentin Berg Professor of Mechanics and Graduate Officer Chairman, Committee for Graduate Students 2 Skin Friction Drag Reduction in Turbulent Flows Using Superhydrophobic Surfaces

by Anoop Rajappan Submitted to the Department of Mechanical Engineering on August 11, 2017, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering

ABSTRACT The use of randomly textured superhydrophobic surfaces have recently gained interest as a potential approach for the passive reduction of skin- friction on the hull of ships, submarines and underwater projectiles. When submerged in water, these surfaces trap a layer of air (or 'plastron') within their texture, which allows the external fluid to partially slip over the bound- ary, decreasing the net frictional shear stress on the wall. Five prototype drag-reducing surfaces were evaluated experimentally as possible candidates for turbulent drag reduction applications, using a combination of flow tests, surface profile measurements, and contact angle goniometry. Three of these were randomly rough superhydrophobic textures, produced by easily scalable mechanical and chemical surface treatment processes; the other two surfaces consisted of periodic streamwise grooves, filled with air or a low viscosity liq- uid lubricant. The skin friction characteristics of all five prototype surfaces were measured in fully turbulent flow inside a custom-built Taylor-Couette apparatus for Reynolds numbers in the range 1.64 x 104 < Re < 8.59 x 104, and a characteristic slip length was determined for each surface from fric- tional torque measurements. Drag reductions of up to 26%, and effective slip lengths as large as 32 pm, were obtained during flow tests on superhy- drophobic surfaces. Consistent with the results of previous studies in the the literature, the percentage drag reduction was found to increase with increas- ing flow Reynolds numbers; furthermore, the effective slip length for each surface remained approximately constant with Re, until the onset of surface failure. The surface morphology of each of the three rough superhydrophobic

3 textures was further characterized through non-contact optical profilometry, and a number of surface statistical parameters quantifying the vertical and lateral length scales of roughness features were computed from the measured height profile. These were then compared with the experimentally deter- mined slip lengths, to extract correlations between specific roughness param- eters and drag reduction performance, and to identify desirable features of surface roughness that enhance slip length and interface stability under tur- bulent flow conditions. Analogous to regularly patterned surfaces, the slip length was found to depend strongly on the lateral spacing of texture as- perities, quantified by the mean autocorrelation length in the case of fractal self-affine rough surfaces. Finally, a large lateral spacing between roughness peaks, along with a small root mean square roughness (compared to the vis- cous length scale of the turbulent flow), and the presence of a hierarchical texture with roughness features at two or more length scales, were identified as the three key requirements for the successful design of rough superhy- drophobic surfaces for turbulent drag reduction in practical applications.

Thesis Supervisor: Gareth H. McKinley Title: School of Engineering Professor of Teaching Innovation

4 Acknowledgments

I express my profound gratitude to Prof. Gareth McKinley, my research ad- visor, for his constant support, guidance and encouragement throughout my research life at MIT. I have greatly admired his enthusiasm for science, his passion for teaching, his objective approach to research and his rigorous at- tention to detail. He has always been understanding and approachable on all matters of research and academic life. I am very fortunate to have had the opportunity to work with him on this project. I thank Divya Panchanathan, fellow graduate student and research col- laborator, for her invaluable support at times of difficulty, and for being there as a friend in times of need. I also thank Alex, Bavand, Michela, Philippe, Shabnam, Caroline, Ahmed, Justin, Dayong and Gibum for making me feel welcome and at home even when I was eight thousand miles away from home. I will always treasure the several happy moments I have had in their company in the last two years. Their friendship has helped me grow both as a person and as a researcher during my time at MIT. I would like to thank Mr. Pierce Hayward for patiently teaching me the basics of machining, and allowing me to use the machines at the Graduate Student Shop, without which I could not have built my experimental appara- tus in time. I thank Mr. Sean Buhrmeseter for his help in all administrative matters and material procurements related to the project. I am also grateful to Ms. Leslie Regan, MechE graduate administrator, for her constant and reassuring support during my initial few months at MIT, and for easing my transition to the challenging research and academic environment here. I am ever grateful to my professors at IIT Madras, my alma mater, where I learned the basics of engineering and took my first steps in research. I espe- cially thank Professors Ashis Sen, K. Ramamurthi and Parag Ravindran, for their valuable guidance and mentorship during my final year at IT Madras while I considered pursuing graduate studies abroad.

5 I acknowledge the contributions of researchers from other universities who collaborated with me in the MURI project, and provided many of the test surfaces that were used in this study. I greatly appreciate the feedback I received from them during open discussions at the MURI review meetings. I thank the Office of Naval Research (ONR) for financially supporting the project. Above all, I am forever indebted to my parents for their unconditional love, their unwavering support in times of crisis, their unending patience and understanding, and their prayers and blessings at every stage of my life; I owe everything to them. I can only hope to give back a fraction of all they have done and sacrificed for my well being. Just three years ago, I wouldn't have dreamt of having an opportunity to do graduate studies at this prestigious institute. I can only attribute it to the grace of the divine providence, for which I am eternally thankful.

6 Contents

1 Introduction 15 1.1 Frictional drag on marine vehicles ...... 15 1.2 Wetting and superhydrophobicity ...... 17 1.3 Slip on superhydrophobic surfaces ...... 22 1.4 Research goals and expected outcomes ...... 26 1.5 Organization of this thesis ...... 27

2 Taylor-Couette Device for Skin Friction Measurements 29 2.1 Introduction ...... 29 2.2 The Taylor-Couette device ... .. 29 2.2.1 Why TC flow?...... 32 2.3 Device design and fabrication .... . 33 2.3.1 The Small TC device .. .. . 35 2.3.2 Upgrading the design - the Large TC device . ... . 38 2.3.3 The AR-G2 rheometer ..... 44 2.4 Flow characterization...... 46 2.4.1 Flow visualization ...... 48 2.4.2 Transition to turbulent flow ...... 52 2.4.3 Torque scaling in turbulent TC flow ...... 56 2.5 C losure ...... 61

3 Characterization of Drag-Reducing Surfaces 63 3.1 Introduction ...... 63 3.2 Experimental methods ...... 66 3.2.1 Skin friction measurements ...... 66 3.2.2 Surface profilometry ...... 70 3.2.3 Contact angle goniometry ...... 75 3.3 Rough superhydrophobic surfaces ...... 75

7 3.3.1 Acrylic FPOSS spray coating ...... 77 3.3.2 Etched aluminum ...... 80 3.3.3 Sandblasted, etched and boehmitized aluminum . . .. 82 3.4 Streamwise grooved surfaces ...... 88 3.4.1 Liquid-infused streamwise grooves ...... 90 3.4.2 Air-filled streamwise grooves ...... 94 3.5 Summary of experimental results ...... 97

4 Conclusion 103 4.1 Designing optimal textures for drag reduction ...... 103 4.2 Summary and outlook ...... 112

A Engineering drawings for the TCL device 115

B Experimental data 121

8 List of Tables

1.1 A list of previous experimental studies on turbulent drag re- duction by rough superhydrophobic surfaces...... 25 2.1 A comparison of the dimensions, geometry ratios and key flow parameters for the TCO, TCS and TCL devices...... 35 2.2 Some previous experimental studies on high Reynolds number T C flow s...... 43 2.3 Technical specifications for the AR-G2 rheometer in steady shear operation...... 45 2.4 Density and viscosity of working fluids used in flow tests in the TCL device...... 52 3.1 List of drag-reducing surfaces used in flow tests and surface characterization studies...... 65 3.2 The angular speeds and the corresponding Reynolds numbers at which frictional torque on the rotor was measured during flow experiments in the TCL device...... 67 3.3 Experimentally measured surface statistics and apparent con- tact angles for rough superhydrophobic test surfaces...... 76 3.4 Physical properties of n-heptane...... 91 3.5 Summary of flow measurements on all drag-reducing test sur- faces...... 100

9 10 List of Figures

1.1 Resistance components of a full form tanker ship in full load and ballast conditions...... 16 1.2 Potential strategies for active and passive skin friction reduc- tion on surface ships, submarines and underwater projectiles. . 17 1.3 A water drop on the leaf of the lotus plant. Scanning electron micrograph of the dorsal surface of the lotus leaf...... 18 1.4 Schematic representations of Wenzel and Cassie regimes. Wa- ter drops of the same volume on a superhydrophobic texture coexisting in Cassie and Wenzel states...... 20

1.5 Flow over a superhydrophobic texture in the Cassie state. . .. 22 2.1 The basic elements of the TC device, and its principal flow dim ensions...... 30 2.2 Sectional view of the TCO device...... 34 2.3 Sectional view of the TCS device...... 36 2.4 The TCS device mounted on the rheometer during a flow test. 37 2.5 Sectional view of the TCL device...... 40 2.6 The TCL device mounted on the AR-G2 rheometer...... 41 2.7 Flow patterns observed in the TCL device at different Reynolds numbers during flow visualization experiments. Part 1. .... 49 2.8 Flow patterns observed in the TCL device at different Reynolds numbers during flow visualization experiments. Part 2. .... 50

11 2.9 Baseline friction curve for the TCL device...... 54 2.10 The local torque exponent as a function of the flow Reynolds num ber...... 55 2.11 Skin friction curve for a smooth rotor in the TCL device, plot- ted in Prandtl-von KArmin coordinates...... 58 2.12 Theoretical prediction for the local torque exponent as a func- tion of the flow Reynolds number Re...... 60 3.1 A depth composed optical micrograph of the acrylic FPOSS spray coating, showing the corpuscular microstructure of the poly(methyl methacrylate) matrix...... 78 3.2 Skin friction data for the acrylic FPOSS spray coating in Prandtl-von Karmain coordinates...... 79 3.3 Percentage DR as a function of the Reynolds number for the acrylic FPOSS spray-coated surface ...... 79

3.4 2D profilogram of the acrylic FPOSS spray coated surface. .. 81 3.5 A depth composed optical micrograph of the etched aluminum surface...... 82 3.6 Skin friction data for the etched aluminum surface in Prandtl- von Kdrmin coordinates...... 83 3.7 Percentage DR as a function of the Reynolds number for the etched aluminum surface ...... 84 3.8 2D profilogram of the etched aluminum surface...... 84 3.9 SEM images of the 80 grit sandblasted, etched and boehmi- tized surface, before and after hydrophobization...... 85 3.10 Skin friction data for the sandblasted, etched and boehmitized aluminum surfaces in Prandtl-von Kdrmin coordinates. .... 87 3.11 Percentage DR as a function of the Reynolds number for the 80 grit and 150 grit sandblasted, etched and boehmitized alu- m inum surfaces...... 88

12 3.12 2D profilograms of the 80 grit and 150 grit sandblasted, etched and boehmitized aluminum surfaces...... 89 3.13 The streamwise grooved rotor after deposition of the organosi- lane monolayer. Drops of water do not wet the rotor surface after the silanization process...... 90 3.14 Skin friction data for the heptane-infused streamwise grooves in Prandtl-von Karmi'n coordinates...... 92 3.15 Percentage DR as a function of the Reynolds number for stream- wise grooves infused with n-heptane...... 92 3.16 The heptane-infused grooved rotor, imaged before and after flow tests. Several grooves have failed by the end of the test, and the displaced heptane is visible as a thin red layer of liquid floating at the top of the cell...... 93 3.17 SEM images of the air-filled streamwise grooves, before appli- cation of the perfluoropolyether lubricant...... 95 3.18 Skin friction data for air-filled streamwise grooves in Prandtl- von Kdrmin coordinates...... 96 3.19 Percentage DR as a function of the Reynolds number for air- filled streamwise grooves, in water and synthetic seawater. .. 97 3.20 The air-filled streamwise grooved surface, imaged before and after flow tests. Air has been displaced from several grooves by the end of the test, especially near the top and bottom ends of the rotor...... 98 3.21 Drag-reducing test surfaces ranked in ascending order of max- imum percentage drag reduction, slip length and wall shear stress at failure...... 99 4.1 Slip length plotted against various roughness parameters for randomly textured superhydrophobic surfaces...... 104 4.2 RMS roughness of superhydrophobic test surfaces in wall units at the point of failure...... 105

13 14 Chapter 1

Introduction

1.1 Frictional drag on marine vehicles

Skin friction accounts for a significant fraction of the total hydrodynamic resistance on surface and sub-surface watercraft; under normal service con- ditions, approximately 50 % of the total drag on ships, and 60 % of the drag on submarines, arises from frictional shear stress on the hull [1]. Figure 1.1 shows the resistance break down of a full form, large block coefficient tanker ship in calm water conditions. Because modern ships have carefully designed hull forms and bulbous bows to mitigate form and wave drag, the net re- sistance at design speed is largely due to skin friction, and approximately 60 % of the propulsive power of a typical displacement ship is expended in overcoming this viscous frictional drag 121. Techniques to reduce skin friction can therefore produce substantial savings in fuel consumption and operating costs, through improvements in ship speed and efficiency; the development of successful drag reduction strategies thus has potential economic, strategic and environmental benefits. Several active and passive methods of reducing skin friction have been explored in the past, with varying levels of success [1]; Figure 1.2 lists some of these drag reduction techniques. Active methods for frictional drag reduc- tion include injection of air bubbles 13] or high molecular weight polymers [4] into the turbulent boundary layer, and incorporating submerged air-filled recesses, or 'partial air cavities', under the hull 151. These methods require some form of continuous input to sustain drag reduction, and become eco- nomically viable in long-haul operations only if the cost involved in active

15 100 &4Full Load

Ship A Lpp= 302m Cb= 0.833

50 - 0.

010

10 12 14 16 Vs (kN)

0.10 0.12 0.14 U /,FIT

Figure 1.1: Resistance components of a full form tanker ship with block coefficient CB = 0.83 in full load and ballast conditions. The block coefficient CB is the ratio of the volume displaced by the hull, to that of a cuboidal box formed by the waterline length, beam and draft. [Reproduced from Larsson, L., and Baba, E., Ship resistance and flow computations, in Advances in marine hydrodynamics, Computational Mechanics Publications, vol. 5 ch. 1, pp. 1-75 (1996).] flow modification is offset by the savings in fuel and operating costs [1]; they may also be suitable in military applications that require increased vessel speed and performance over relatively short distances. Passive drag reduc- tion methods, on the other hand, do not require any form of continuous energy input, but usually produce a smaller percentage reduction in drag than active methods. For example, shark-skin inspired riblet surfaces have been extensively investigated in the literature, and are reported to reduce wall shear stress in turbulent boundary layer flows by modifying the near- wall vortex dynamics [6]. Recently, superhydrophobic (SH) coatings on the

16 Skin friction drag reduction Active methods - Polymer injection (PDR)

Air bubble injection (ALDR)

L Partial air cavities (PCDR)

Passive methods Partial- and super-cavitation

H Streamwise riblets L Superhydrophobic surfaces

Figure 1.2: Potential strategies for active and passive skin friction reduction on surface ships, submarines and underwater projectiles. Abbreviations - PDR: poly- mer drag reduction, ALDR: air layer drag reduction, and PCDR: partial cavity drag reduction.

hull have gained interest as a potential technique for skin friction reduction on ships and undersea vehicles [1]. However, scalable SH surfaces that can successfully operate at the high Reynolds numbers (Re ~ 108 - 109) typically encountered in marine applications, are yet to be developed [7]. The focus of this study is on the experimental characterization of randomly rough super- hydrophobic textures, as candidate surfaces for passive skin friction reduction in wall-bounded turbulent flows.

1.2 Wetting and superhydrophobicity

A drop of water deposited on the leaf of the lotus plant Nelumbo nucifera rolls off without wetting the surface [8]. A closer look at the dorsal side of the leaf [Figure 1.3(b)] reveals the presence of a complex hierarchical roughness, consisting of microscopic bumps or papillae 10-50 pm in size, covered by a fine nanotexture of epicuticular wax crystalloids at the scale of about 100 nm [9]. The extreme water repellency and self-cleaning ability of the leaf are the result of a thin layer of air retained between the asperities, shielding most of the leaf surface from direct contact with the water drop [10]. This layer of entrapped air, termed plastron, is visible as the silvery sheen underneath the' water drop in Figure 1.3(a); the luster arises due to total internal reflection

17 I

Figure 1.3: (a) A water drop does not wet the leaf of the lotus plant Nelumbo nucifera. The silver sheen visible under the drop arises from total internal reflection of light at interface of the trapped air layer, or plastron. (b) Scanning electron micrograph of the dorsal surface of the lotus leaf; the scale bar denotes 20 Pm. [Reproduced from Barthlott, W., and Neinhuis, C., Purity of the sacred lotus, or escape from contamination in biological surfaces, Planta 202(1), 1-8 (1997).] of ambient light at the air-water interface [11]. Surfaces such as the lotus leaf, that display extremely high water repellency and very low drop stiction, are called superhydrophobic surfaces [10]. Nature provides several examples of superhydrophobic surfaces - the leaves of many plants, including the lotus, taro, broccoli and kale [12], the wings of insects such as butterflies and cicadas [17,18], the compound eyes of the common mosquito [20], and the feathers of birds such as ducks, geese, pigeons and penguins 113-15], are all found to exhibit superhydrophobicity. Water striders have superhydrophobic legs covered with tiny hairs or mi- crosetae, enabling them to float and skate on water with ease [191. Several aquatic insects rely on the plastron layer trapped by their superhydrophobic body surface to breathe underwater without gills [16]. A careful examina- tion of these natural surfaces reveals two factors that contribute to their superhydrophobicity - a low surface energy, usually achieved by a coating of epicuticular wax, and a rough microtexture, often with hierarchical features at multiple length scales. A liquid deposited on a solid surface either spreads or coalesces into drops; the wetting behavior on an ideal surface is determined by the sign of the spreading coefficient S, defined as [10]

S = USA - (USL + LA), (1-1)

18 Here USA, USL and ULA are, respectively, the surface tensions of the solid-air, solid-water and water-air interfaces. For S > 0, the liquid spreads sponta- neously as a film and fully wets the solid. For S < 0, the liquid forms a spherical lens on the surface, with a finite contact angle 0 at the base. For an ideal solid surface that is smooth and chemically homogeneous, the contact angle Oy at equilibrium is given by the Young equation 121]:

cos Oy = SA - SL (1.2) ULA

The angle Qy is variously called the equilibrium contact angle, the Young angle or the chemical contact angle. The Young equation shows that on a surface free of roughness and chemical inhomogeneities, the equilibrium angle at the three-phase contact line is uniquely fixed by the chemical nature of the intersecting phases. In the case of water, Oy varies from 0' on high-energy solids, such as clean glass and metal surfaces, to a maximum of about 1200, on self-assembled monolayers of highly fluorinated alkanoic acids [221. A solid substrate is termed hydrophilic if Qy < 900, and hydrophobic if Qy > 900. Real surfaces, however, are seldom smooth, and the presence of roughness has a profound influence on the wetting behavior of liquids on solid surfaces [101. Wenzel (1936) was the first to quantify the effect of surface roughness on the apparent contact angle 0*, the average macroscopic angle measurable at the contact line [23]. The Wenzel model assumes that when the contact line advances on the dry substrate, the liquid conformally wets all roughness features, penetrating the surface texture and fully displacing the gas phase; the apparent angle is then given by

cos 0* = rw cos Oy (1.3)

The parameter rw, called the Wenzel roughness, is the ratio of the actual (or developed) area of the surface to its projected area. Because rw > 1, the Wenzel equation predicts that roughness simply enhances the intrinsic wet- tability of the solid - a hydrophilic surface, when roughened, becomes more hydrophilic (0* < 9y), and a hydrophobic surface becomes more hydrophobic (9* > Oy). Although these trends are qualitatively observed, the unique ap- parent angle given by equation (1.3) is rarely measured during experiments; instead, a range of metastable apparent angles can be observed in practice, due to pinning of.the contact line on roughness asperities and surface defects [10]. The largest value of the apparent contact angle, observed when the

19 a regularly patterned superhydrophobic surface. (c) Millimeter-size water drops of the same volume on a superhydrophobic texture consisting of regularly spaced posts (# = 0.01). The left drop is in a metastable Cassie state, as evidenced by light passing under it. The right drop was gently pressed, which induced a transition to the more stable Wenzel state. (Reproduced from Callies, M., and Qu.r., D., On water repellency, Soft Matter 1, 55-61 (2005) .. contact line advances on the dry substrate, is called the advancing contact angle 1. [24. Similarly, the smallest value of the apparent contact angle, measured as the contact line recedes from the previously wetted substrate, is called the receding contact angle 9r. The difference between the advancing and receding angles is termed the contact angle hysteresis, Mlme = ( 9 -wtr). A small value of lm indicates high droplet mobility; this inference follows from the expression for the critical force F necessary to initiate motion of a liquid drop of basal radius r on the surface [25 :

F = srrzL A (cOd - coOsa) (1.4)

In the Wenzel regime, the surface texture is fully inundated by the liquid [Figure 1.4(a) , resulting in strong contact line pinning and large contact

20 angle hysteresis; as a result, water droplets in the Wenzel state stick to the surface and do not slide easily under gravity, even when the underlying solid is hydrophobic. On the other hand, if the hydrophobic substrate is sufficiently rough, a second wetting regime called the Cassie state is observed 110]. In this case, as the contact line advances, the liquid simply bridges across surface asperities, leaving pockets of air confined between the solid surface and the newly formed liquid-air interface [Figure 1.4(b)]. Assuming that the liquid meniscus remains largely flat, the apparent contact angle in this composite wetting regime is given by [261:

cos 0* = -1 + (r cos Oy + 1) (1.5)

Here, the parameter # is the areal fraction of the liquid interface in direct contact with the solid substrate, and ro is the Wenzel roughness of the wet solid region. With a large fraction of the surface covered by air, direct contact between the liquid phase and the solid substrate is minimal, and contact line pinning is drastically reduced. Consequently, the Cassie wetting regime is characterized by large apparent contact angles and very low contact angle hysteresis, and thus corresponds to the true superhydrophobic state [10]. Even on moderately rough hydrophobic substrates, the Cassie regime is quite frequently observed in practice, although the Wenzel state has a lower interfacial free energy and represents the thermodynamically stable configu- ration of the system [10]. These 'metastable' Cassie states occur when the initial penetration of the liquid into the surface texture is impeded by an en- ergy barrier arising from contact line pinning, large aspect ratio asperities, or re-entrant overhangs [27-30]. Air inside the texture persists in a metastable state, until an irreversible transition to the Wenzel regime is triggered by external pressure, mechanical vibration or the diffusive loss of gas into the liquid phase [31-331. Once the liquid invades the texture, the resulting Wen- zel state is energetically stable, and the superhydrophobicity of the surface is irrevocably lost. Figure 1.4(c) shows two water drops of the same vol- ume, coexisting in two different wetting regimes on the same microtextured substrate. Aided by the development of precise microfabrication techniques, re- searchers have, over the last two decades, devised an impressive array of synthetic microtextures that emulate the features of many naturally occur- ring superhydrophobic surfaces. Artificial surfaces investigated over the years

21 1 2 3

+

Figure 1.5: Flow over a superhydrophobic texture in the Cassie state. The solid asperities act as no-slip boundaries (1) and the liquid-air interfaces act as shear-free boundaries (2); the flow far from the surface is approximately modeled by replacing the composite boundary with an equivalent slip length b at the wall (3).

include highly ordered patterns of ridges or posts fabricated by microlithogra- phy, fiber bundles formed by electrospinning, and randomly textured surfaces produced by a variety of techniques such as spray-coating, chemical etching, sandblasting, thermal deposition and boehnitization 134-401. A number of anti-rain and water-repellant superhydrophobic treatments are now commer- cially available for industrial and household applications.

1.3 Slip on superhydrophobic surfaces

The air layer, or plastron, trapped within the surface texture in the Cassie state, can induce slip and reduce frictional drag in flow over superhydropho- bic surfaces 1411. Emperor penguins, for instance, are known to utilize the air trapped in their plumage to reduce drag as they ascend from underwater dives [15]. Consider the case of simple viscous shear flow over a superhy- drophobic texture, as shown in Figure 1.5. In the regions where the liquid is in direct contact with the solid substrate, the usual no slip boundary con- dition applies. By contrast, the air-water interfaces bridging the asperities of the texture act effectively as shear-free boundaries, since the viscosity of air is considerably smaller than that of water (by a factor of almost 50 at room temperature). Consequently, the flow partly slips over the surface, and the viscous shear stress on the wall is reduced because of the diminished

22 contact between the solid substrate and the flow. The heterogeneous bound- ary condition on the superhydrophobic texture is modeled by prescribing an equivalent slip velocity at the wall, given by the Navier slip condition 142]:

us = bx T = bx - vs = by z = by -- ws = 0 (1.6) P dz ZO A dz ,__O Here, x, y and z represent the streamwise, spanwise and wall-normal coor- dinates, u(z), v(z) and w(z) are the flow velocities, [ is the viscosity of the liquid, and T2x and Tz are components of the wall shear stress in the x and y directions. The nominal plane of the wall is at z = 0, and we have further assumed that the flow is fully developed in the streamwsie and spanwise di- rections. The slip velocities us and vs are respectively proportional to the effective slip lengths bx and by on the surface; on a rough superhydrophobic texture, we expect the slip to be isotropic, giving a single characteristic slip length bx = by = b. Drag reduction over a superhydrophobic surface was first demonstrated experimentally by Ou et al. (2004), for laminar flow in a microchannel with one wall textured with regular arrays of microridges and microposts 1431. Ou and Rothstein (2005) used 1 -PIV to measure velocity profiles for the laminar flow over regularly patterned streamwise ridges in a inicrochannel 1441; they obtained slip lengths of the order of the size of the surface features, and observed that drag reduction increases with decreasing solid fraction #. Lee and Kim (2008) used deep reactive ion etching (DRIE) to fabricate arrays of silicon posts and ridges with remarkably low solid fractions (0 = 0.03 % and 2 % respectively); to stabilize the meniscus between the widely spaced features, the sides of the posts and ridges were nanotextured using a metal- assisted HF-H 202 etching process 1451. Slip lengths as large as 140 pm for the post array, and 400 pm for the ridges, were reported from measurements on a rheometer. Philip (1972) derived analytical expressions for the effective slip length in Stokes flow over periodic arrays of longitudinal and transverse shear-free slots [46,471; his results were further extended to more flow configurations by Lauga and Stone (2003) 148], Cottin-Bizonne et al. (2004) 149] and Sch6necker et al. (2014) 150]. These theoretical studies show that, at least in the limit of creeping flow, the dimensional slip length b is a function of the spatial periodicity of the boundary condition, but is otherwise independent of flow dimensions. Furthermore, the slip length in the case of longitudinal shear-free slots was exactly twice that of the transverse slots.

23 Ybert et al. (2007) derived generic scaling laws for the slip length on superhydrophobic surfaces, in the limit of zero Reynolds number 1511. In their analysis, the liquid interface on the texture is assumed to be perfectly flat and shear-free. The friction on the surface then arises solely from regions

of solid-liquid contact, and therefore T - #p, where 1 is the local shear rate on the solid surface. In the Stokes flow regime, the shear rate scales as ~ U/a, where U is the bulk velocity of the flow and a is the typical length scale of the contact spots; this yields r ~ #pU/a. In the limit of vanishing solid fraction, the velocity profile near the wall approaches that of a plug flow, and the slip length u, ~ U, the bulk flow velocity. Then by definition, the slip length is given by b =us/T, which gives the scaling relation b ~ a/0. For periodic two-dimensional grooves, # = a/L, where L is the periodicity of the groove pattern; the scaling law then simply reads b ~ L, predicting that the slip length is independent of q to the leading order, in agreement with the exact solution which only has a logarithmic dependence on 0. For a two-dimensional array of posts, # ~ (a/L)2 , and the expected scaling for slip length is b ~ L/VT, which agrees well with their computational results. Superhydrophobic drag reduction in turbulent flows brings several new challenges not present in the laminar case 17]. Large velocity and pressure fluctuations in strongly turbulent flow can trigger a wetting transition to the Wenzel state, after which the surface texture simply acts as hydrodynamic roughness, increasing frictional drag. The viscous sublayer and near wall co- herent structures can interact in complex ways with the roughness elements of the texture, and potentially offset any friction reduction arising from wall slip. Min and Kim (2004) performed DNS studies of turbulent channel flow, with prescribed steamwise or spanwise slip boundary conditions at the wall [521. Whereas streamwise slip resulted in drag reduction, spanwise slip led to strengthening of near wall vortex structures, enhancing turbulent momentum transport and increasing the frictional shear stress on the wall. This mecha- nism of drag increase is unique to turbulent flows, and is absent in laminar flows. On random textures which have equal slip lengths in both stream- wise and spanwise directions, drag reduction generated by streamwise slip is therefore partly offset by the drag increase due to spanwise slip. Martell et al. (2010) arrived at similar conclusions regarding the effect of streamwise and spanwise slip [53], and in a separate study showed that drag reduction in turbulent flow increases with Reynolds number, unlike in the case of lam- inar flow [54]. Recently, Seo and Mani (2016) derived a scaling law for slip length valid in the high Reynolds number limit, and validated it using DNS

24 Table 1.1: A list of previous experimental studies on turbulent drag reduction by rough superhydrophobic surfaces, along with the flow setup utilized for measure- ments, the range of shear Reynolds numbers (Re*) investigated, and the percentage change in frictional drag (% DR) observed. Negative values of % DR indicate drag increase in the presence of the superhydrophobic texture. [Table adapted from Golovin, K. B., Gose, J. W., Perlin, M., Ceccio, S. L., and Tuteja, A., Bioinspired surfaces for turbulent drag reduction, Philosophical Transactions of the Royal So- ciety A 374, 20160189 (2016).]

Reference Flow setup Re* range % DR [56] Gogte et al. (2005) Water tunnel 40 to 288 +3 to +18 [57] Zhao et al. (2007) Water tunnel 1700 to 3300 -5 to +9 [581 Peguero and Breuer (2009) PIV 200 -50 to +40 [591 Aljallis et al. (2013) Towing tank 520 to 5170 -30 to +30 [401 Bidkar et al. (2014) Water tunnel 1000 to 5000 -13 to +30 [60] Srinivasan et al. (2015) TC flow 480 to 3810 0 to +22 [611 Zhang et al. (2015) PIV 329 to 467 +10 to +24

results [55]. In their analysis, a laminar boundary layer is assumed to de- velop over each solid-liquid contact spot, and the Stokes scaling T ~ #pU/a is replaced by the boundary layer scaling T ~ #Pu,/J, where 6 ~ i'/a/u, is the thickness of the boundary layer. The scaling law for slip then becomes b+ ~ (L+)1/ 3 0-1/ 2 , where L+ and b+ are, respectively, the surface periodicity and the slip length scaled in wall units. Unlike the result for low Reynolds numbers, the dimensional slip length b = L1/32/ 3 -1/2 is now also a function of the viscous length scale 6,. Daniello et al. (2009) obtained drag reduction in turbulent microchannel flow over superhydrophobic streamwise ridges fabricated from polydimethyl siloxane [621. The measured drag reduction approached the theoretical max- imum of (1 - #), and slip lengths over 100 pm were reported at the largest Reynolds numbers. In contrast to regularly patterned textures, the avail- able experimental data for turbulent drag reduction on random textures has mostly been inconclusive, and in some cases, contradictory; drag reduction as well as drag increase have both been reported on similar surface textures [7,59,61,631. The disparity in experimental results are probably due to dif- ferences in surface preparation and flow conditions, which vary between test facilities. Unlike the case of regular arrays of ridges and posts for which

25 the wetting transition is abrupt, the liquid-air interface on a random tex- ture traverses several intermediate metastable configurations with increasing external pressure 164]; surface failure occurs progressively and may not be immediately apparent. As the interface penetrates deeper into the texture, the wetted solid fraction increases, altering the effective slip length; conse- quently, identical rough textures under different flow conditions can yield widely different values of slip length. Table 1.1 lists a selection of past ex- perimental studies on drag reduction by randomly rough superhydrophobic surfaces in turbulent flow.

1.4 Research goals and expected outcomes

The present study was conducted as part of a wider research effort to in- vestigate passive skin-friction reduction using superhydrophobic surfaces in turbulent boundary layer flows. Teams from several universities collaborated in a Multidisciplinary University Research Initiative (MURI), funded by the Office of Naval Research (ONR) of the US Department of Defense. The aim of the MURI project was to develop a mechanistic understanding of tur- bulent drag reduction by superhydrophobic textures, and to design practical drag-reducing surfaces for real-life service with emphasis on naval vessels and underwater projectiles. The research presented in this thesis was undertaken with three principal objectives in mind: 1. Design a simple and reliable experimental setup for measuring the slip length and percentage skin friction reduction on prototype drag- reducing surfaces in high Reynolds number turbulent flow; 2. Characterize the drag reduction performance and surface morphology of scalable, randomly textured SH surfaces, developed at MIT and at other universities by collaborating research groups in the MURI project; and, 3. Study how different statistical parameters of the surface profile affect the slip length and interface stability on superhydrophobic surfaces, and deduce design rules for fabricating optimal textures for turbulent drag reduction. Accordingly, a custom Taylor-Couette (TC) fixture, modeled after the experimental apparatus used by Srinivasan et al. (2015), was built to per- form skin-friction measurements on candidate drag-reducing surfaces [60].

26 The original design of the apparatus was upgraded, and the critical rota- tion rate for transition to full turbulence in the new device was determined experimentally from baseline tests. Afterwards, friction measurements were performed on five different test surfaces, of which four were provided by col- laborating research groups in the MURI project. Three of the surfaces tested were rough superhydrophobic textures, and the topological features of these surfaces were further characterized through non-contact optical profilome- try and contact angle measurements. Finally, experimentally measured slip lengths and surface statistics were compared to draw connections between specific roughness parameters and drag reduction performance.

1.5 Organization of this thesis

The first chapter states the key research objectives and the expected out- comes of the study; also included is a brief survey of current literature on drag reduction by superhydrophobic surfaces in laminar and turbulent flows. Chapter 2 describes the design and baseline characterization of the Taylor- Couette apparatus used in all friction measurements on drag-reducing test surfaces reported in this thesis. Chapter 3 discusses the fabrication and experimental evaluation of five prototype drag-reducing surfaces; measured slip lengths and profile statistics are reported for three different rough super- hydrophobic textures. Chapter 4 concludes the thesis with a set of design guidelines for fabricating optimal rough textures for turbulent drag reduc- tion, and a short discussion on possible avenues for future research.

Supplementary information. Engineering drawings for the TCL device de- scribed in chapter 2 are included as appendix A. Raw experimental data from flow measurements on all test surfaces are listed in full in appendix B.

27 28 Chapter 2

Taylor-Couette Device for Skin Friction Measurements

2.1 Introduction

To investigate the drag reduction performance of superhydrophobic surfaces, we need a simple and reliable experimental technique to measure the skin friction and slip length on different test surfaces under turbulent flow condi- tions. Techniques such as pipe flows, water tunnels, tow tank tests and PIV have been employed in previous investigations of slip on drag reducing su- perhydrophobic surfaces. In this study, the wall shear stress on test surfaces was measured inside a custom-built Taylor-Couette (TC) device, operated at rotation rates that are sufficiently large to generate shear-driven turbulence in the flow gap. The subsequent sections of this chapter describe how the TC apparatus was designed and fabricated, the critical Reynolds number for transition to turbulence was identified, and the baseline friction curve in the turbulent flow regime was determined.

2.2 The Taylor-Couette device

The Taylor-Couette (or 'TC') device, in essence, comprises two concentric circular cylinders - a rotating inner cylinder, hereafter called the rotor, and a stationary outer one, called the stator. Figure 2.1 shows schematically, the apparatus and its key components. The annulus between the two cylinders is filled with water, and the rotating inner cylinder sets up a shear driven flow

29 Torque T Angular speed Q

d

L Stator

Gap

Rotor

Figure 2.1: The basic elements of the TC device, and its principal flow dimensions. Fluid is confined in the gap between two concentric cylinders - the outer cylinder remains stationary and is called the stator, whereas the inner cylinder can rotate freely and is called the rotor. in this gap. Due to the no-slip condition at the stator wall, the azimuthal velocity and angular momentum of the fluid inside the gap decreases with radius. As a result, the laminar, purely circumferential, primary flow is unstable to radial perturbations beyond a critical angular speed of the rotor; when the speed is increased above this value, the flow passes through a number of states, and different kinds of secondary vortical flows appear and become stable [65-69]. If the rotational speed of the rotor is increased even further, these secondary flow structures become turbulent and eventually dissipate, until finally they cannot be detected any longer; at this stage, shear-driven, featureless turbulence has been attained in the gap [68,70,71]. The Taylor-Couette geometry thus provides a simple way to generate and study turbulent shear flows inside a compact flow setup. A Taylor-Couette cell is characterized by three key dimensions: the inner (or rotor) radius Ri, the outer (or stator) radius R,, and the height of the flow cell L; these dimensions are marked in figure 2.1. The radial gap between

30 the stator and the rotor is d = R, - Ri. We next define three important non-dimensional geometry ratios:

= , radius ratio RO d gap ratio - and (2.2) Rj) L aspect ratio = . (2.3) d If Q denotes the angular speed of the rotor, the azimuthal speed of the rotor surface is V = RjQ. Let v be the kinematic viscosity of the liquid inside the gap. The flow Reynolds number Re is defined using V and d as the velocity and length scales: Re = =-QRd (2.4) Another dimensionless group relevant to TC flows is the Taylor number Ta:

2 Ta = A Ri+R 1 d2Q2 (2.5) 1 2 1 V2 The factor A is a constant defined in terms of the radius ratio q of the Taylor-Couette system:

A = q) (2.6) 16,q2 This definition of the Taylor number, suggested by Eckhardt, Grossmann and Lohse (2007), is based on an exact analogy between the equations governing Taylor-Couette flow and Rayleigh-B6nard (RB) convection [72]. The Taylor number Ta is analogous to the Rayleigh number Ra, and the factor A is a 'geometric' Prandtl number for TC flows, analogous to the fluid Prandtl number Pr in RB convection. Rearranging equation (2.5) shows that the Reynolds number and the Taylor number are related:

Ta = (1 Re 2 (2.7) 64 94 In the present study, the TC apparatus was chosen over other experimen- tal methods for skin friction measurement, on account of the simplicity of the flow geometry, the ease of building and operating the device, and a few other

31 practical considerations. The cylindrical symmetry of the flow, for instance, permits a simple and straightforward determination of skin friction and slip length by measuring the steady-state torque on the rotor. The key reasons for selecting the TC system are briefly discussed in the next section.

2.2.1 Why TC flow? In the TC device, the superhydrophobic texture to be tested is applied to the outer surface of the rotor. Provided the operating speed is high enough for secondary flow structures to have disappeared, the shear stress T, and therefore the shear Reynolds number Re*, are uniform over the cylindrical rotor surface.1 Hence, all regions of the test surface experience uniform turbulent flow conditions - the same turbulence intensity, Reynolds stresses and magnitude of pressure fluctuations - since these depend only on the magnitude of the shear Reynolds number Re*. In contrast, the local shear stress on a flat plate changes along the flow direction in a water tunnel or tow tank experiment, as the boundary layer develops from the leading edge to the trailing edge of the plate. The wall shear stress measured in a TC experiment corresponds to a homogeneous local shear stress acting over the entire test surface, and is not the area-averaged value for a spatially varying shear stress field. Wall-bounded turbulence with uniform wall shear stress can also be ob- tained in fully developed internal (or pipe) flows; however, in these systems, a driving pressure gradient in the flow direction is also required. The test section, therefore, has to be kept short so that the fluid pressure over the superhydrophobic texture is approximately uniform. This is particularly im- portant when testing randomly rough superhydrophobic textures, on which the liquid-air interface can occupy many different metastable, partially wet- ting configurations, depending on the applied external pressure. If the fluid pressure changes appreciably along the length of test section, the degree of interface penetration, the extent of solid-liquid contact, as well as the local slip length, will all vary from one location to another on the surface. In a TC flow, this problem is altogether avoided because the pressure gradient is in the radial rather than the streamwise (or azimuthal) direction; barring tur- bulent fluctuations and the effect of gravity (both present in real-life service

'The shear Reynolds number Re*, and other non-dimensional groups in wall bounded turbulence, are defined in 2.4.3

32 conditions), we may expect the surface of the rotor to be at a uniform static pressure. Another benefit of using TC flow for measuring skin friction resistance is the complete absence of the pressure resistance component (or form drag). During steady rotation, the torque on the rotor arises solely from the shear stress exerted by the fluid; the cylindrical geometry inherently precludes any contribution to the torque from stresses acting normal to the wall.2 This means that a direct measurement of wall shear stress is possible, unlike in a tow tank experiment in which the magnitude of pressure resistance has to be estimated and deducted from the total resistance, to obtain the contribution from shear stress on the wall. In a stand-alone Taylor-Couette flow facility, the end bearings and hy- draulic seals can prove quite challenging to design; these components must operate with minimal friction, add no noise to the rotor torque measurement, and also maintain a leak-proof liquid seal. Further, accurate calibration of angular speed and torque transducers is crucial to obtaining reliable exper- imental data. In our case, the easiest recourse was to take advantage of a commercial rheometer available in the lab. The rotor is attached to the spin- dle of the rheometer, which is supported on a frictionless magnetic bearing that allows speed and torque to be measured with precision. Within the TC device itself, friction is minimized by avoiding all direct surface-to-surface contact between rotating and stationary parts. The exact design and con- struction of the apparatus is explained in more detail in subsequent sections.

2.3 Device design and fabrication

The Taylor-Couette device used in this study is an upgraded version of the wide-gap TC apparatus designed by Srinivasan et al., and described in their 2015 research paper on drag reduction by a spray-coated superhydrophobic surface in TC flow [60]. A sectional drawing of their original device is shown in Figure 2.2; the key dimensions, geometry ratios and flow parameters are listed in Table 2.1. The base and the lid are aluminum, whereas the stator

2In this context, we use the terms 'normal stress' and 'form drag' from the perspective of the macroscopic flow, ignoring the microscopic details of the superhydrophobic texture on the rotor surface. From a microscopic vantage point, the form drag on the micro- asperities protruding into the flow does contribute to the 'macroscopic' shear stress T acting tangential to the nominal profile of the wall 1401.

33 6 20 mm 4

2 5 3 F7

I

.7

Figure 2.2: Sectional view of the TCO device. The labeled parts are: (1) stator, (2) cell gap, (3) rotor, (4) lid, (5) base, (6) sealer ring, (7) circular recess for the Peltier plate and (8) rotor end-recess. is made of glass; this allows the flow inside the gap to be viewed or imaged during experiments, and the plastron to be monitored in real time. The ends of the glass stator fit into recesses in the lid and the base, and o-rings are used to seal the joints against leaks. The base fits securely onto the Peltier plate of the rheometer, and ensures that the stator is concentric to the rotor when the cell is assembled. The rotor, made of aluminum, is treated with the superhydrophobic sur- face to be tested, and attached to the spindle of the rheometer. It is then lowered vertically into the water-filled stator through the circular hole in the lid, the size of which is just enough to let the rotor pass through. When immersed, the end-recess of the rotor traps a pocket of air inside, providing a shear-free interface and avoiding torque contributions from the bottom face of the rotor. After the rotor is in position inside the stator, the lid opening is sealed using a thin acrylic sealer ring, and high-vacuum grease is applied to

34 Table 2.1: A comparison of the dimensions, geometry ratios and key flow parame- ters for the three TC devices. These values are for a smooth untreated rotor, with water as the working fluid. Asterisks denote rough estimates.

Parameter Unit TCO TCS TCL

DIMENSIONS Rotor radius Ri mm 14.0 14.0 38.1 Stator radius R, mm 34.3 34.0 50.8 Rotor length L mm 42.0 42.0 76.2 Cell gap d mm 20.3 20.0 12.7 Bottom end gap mm 2.50 3.00 0.10

GEOMETRY RATIOS Radius ratio 7 0.41 0.41 0.75 Gap ratio E 1.45 1.43 0.33 Aspect ratio 2.07 2.10 6.00

FLOW PARAMETERS Max. Reynolds number Re 79600 78400 86800 Transition Re number Ret 10000 10000 11 000 Max. angular speed Q rad s- 1 250 250 160 Max. peripheral speed V, ms-1 3.50 3.50 6.10 Max. shear Re number* Re* 3800 3800 3000 Max. shear stress* T Pa 25 25 50 Transition viscous length* 6, Jim 27 27 18 Min. viscous length 6 im 5 5 4 hold it in place and prevent leaks. A small clearance, approximately 1 mm, is left between the sealer ring and the rotor stem so that the latter can spin freely. The shorthand 'TCO' is used hereafter to denote this original TC device.

2.3.1 The Small TC device The first TC apparatus built for preliminary tests on superhydrophobic sur- faces was a slightly modified variant of the original TCO device; it had the same stator and rotor dimensions, but included a few design changes and improvements. We shall refer to it as the 'Small TC' device, or 'TCS' for short. This new device was fully transparent, with all parts machined from

35 lo 20 mm

5

6 4

10 10

9

Figure 2.3: Sectional view of the TCS device. The labeled parts are: (1) stator, (2) cell gap, (3) rotor, (4) viewing box, (5) lid, (6) base, (7) liquid seal assembly, (8) rotor end-recess, (9) circular recess for the Peltier plate and (10) fluid inlets. optically clear cast acrylic and bonded together using clear acrylic cement. Acrylic was chosen for its optical clarity, and cast acrylic was preferred to extruded acrylic because of its superior surface finish, machinability and re- sistance to crazing. The ends of the stator were fit into circular grooves carefully milled on the lid and base, to ensure it was concentric with the rotor after assembly. In the TCO device, the o-rings would sometimes fail to prevent leaks, especially at high rotational speeds. To avoid this prob- lem, the stator was permanently solvent welded to the lid and base using dichloromethane (Scigrip 4 acrylic cement) to form transparent, watertight joints. As in the TCO design, a circular recess on the base plate was used to mount the whole assembly on the Peltier plate of the rheometer. The rotors used in TCS were identical to the ones used in TCO, with the same recessed-end design to trap air and create a shear-free interface at the bottom. These were turned on a manual lathe, from aluminum 6061-T6

36 Figure 2.4: The TCS device mounted on the rheometer during a flow test. The smooth aluminum rotor is visible through the viewing box.

rod stock purchased from McMaster-Carr Supply Company, Princeton, NJ. The lid seal, however, was completely redesigned; this was because vacuum grease could not always be relied on to hold the sealer ring in place, and occasional failures would result in air bubbles entering the flow at high rotor speeds. Further, positioning the sealer ring by hand, repeatably, before each experiment was difficult, and if excess grease was applied, there was also the risk of fouling the test surface while lowering the rotor through the lid. In the new design, the acrylic sealer ring includes a thin steel band, and is held in place inside a circular recess by a ring magnet embedded in the lid. As the rotor is lowered, excess water overflows through the lid opening and some of it collects in the recess, submerging the sealer ring and completely sealing the cell against air ingress. The use of vacuum grease is thus avoided altogether.

37 In addition to these modifications, the TCS device had two new features not present in the original TCO design. The first of these was a square viewing box surrounding the stator, with plane transparent walls made of optically clear cast acrylic. Before imaging the cell during experiments, the space between the stator and the outer box was filled with glycerol, a liquid with refractive index comparable to acrylic.3 The viewing box with its flat outer faces reduces the severe cylindrical distortion otherwise produced by the curved stator wall, improving the quality of images obtained. The other new feature was a pair of fluid inlets through which dye or reagent solutions could be introduced into the flow during experiments. Starting outside the device, each inlet channel runs horizontally through the base plate and opens upwards into the cell gap, approximately midway between the stator wall and the rotor; a ball valve is provided at the outer end to shut off each inlet when not in use. Figure 2.3 shows a sectional view of the device, with the key components labeled. The TCS device has the same flow cell dimensions and range of operation as the TCO device; these are again listed in Table 2.1. Figure 2.4 shows the device mounted on the rheometer during an experimental run.

2.3.2 Upgrading the design - the Large TC device The TCO and TCS devices were built as 'wide-gap' cells with a gap ratio of 1.43, much larger than the typical values of E < 0.5 used in most Taylor- Couette flow experiments; this was done to attain transition to turbulence at a smaller critical Reynolds number, so that a wide range of flow Reynolds numbers could be investigated. The radius ratio q = 0.41 of these devices is also uncharacteristically small for a Taylor-Couette apparatus; nearly all available experimental data on TC flows are for much larger radius ratios, usually 0.60 and above [68,70,71,73,741. Although data obtained from TCO and TCS are qualitatively consistent with previous experimental studies in the literature, a direct, quantitative comparison could not be made because of the widely different radius ratios. Another point of concern with these devices is their 'stocky' design; with an aspect ratio of 2.1, the height of the rotor is only about twice the width of the gap. In all TC devices used in this study, the top and bottom walls of the flow cell are fixed to the stator

3 The refractive index of glycerol is 1.47, compared to 1.49 for acrylic (at 589.3 nm, the mean wavelength of the yellow D-lines of sodium).

38 and do not rotate; as a result, B6dewadt layers develop at these walls, where the slow-moving fluid is driven radially inwards by the pressure gradient of the primary azimuthal flow [68]. The radial circulation induced by these sec- ondary flows persist at large Reynolds numbers even after the Taylor vortices have disappeared; its effect on the rotor torque is difficult to quantify, but is usually small and is ignored in the calculation of wall shear stress. A large aspect ratio is therefore important to ensure that end effects in torque mea- surements are indeed minimal; high-end Taylor-Couette facilities typically have aspect ratios ( > 10, and use a segmented inner cylinder with the top and bottom sections isolated from the torque sensor. To address these drawbacks, the dimensions of the flow cell were reworked to improve the geometry ratios of the flow cell, and extend the range of operation to higher wall shear stresses and turbulence intensities. The goal was to design an upgraded version of the TC apparatus that could reach higher peripheral speeds and shear stresses, without exceeding the torque and speed limits of the rheometer. The device had to fit within the available clearance between the rheometer head and Peltier plate, restricting overall dimensions to a maximum of 6 x 6 x 6 inches. The total weight of the device, with the stator and viewing box filled, had to be kept as low as possible to avoid damaging the sensitive 50 N normal force transducer underneath the Peltier plate. Similarly, the rotor had to be designed to minimize the pendant weight in air, and the buoyant upthrust on the rheometer spindle when immersed in water. The dimensions of the flow cell were finalized taking into account these constraints, and a new version of the TC device was built as per the revised design. All slip lengths and skin friction curves reported in this study were measured in this upgraded device, which we refer to, hereafter, as the 'Large TC' device, or 'TCL'. Figure 2.5 shows a sectional drawing of the TCL device and its main components; a full engineering drawing can be found in appendix A. The key dimensions, geometry ratios, and range of operation are listed in Table 2.1. The TCL device was designed with larger stator and rotor radii, and a smaller cell gap, than the TCO and TCS devices; the rotor radius was increased primarily to obtain higher surface speeds and wall shear stresses over the same range of rotation rates. A large radius also helps mitigate the effect of surface curvature on spray-coating, acid etching and sandblasting processes used to generate the superhydrophobic texture on the rotor surface; these texturing processes can be sensitive to substrate curvature, and may yield slightly different results on flat and curved surfaces, even under identical

39 7

9

Figure 2.5: Sectional view of the TCL device. The labeled parts are: (1) stator, (2) cell gap, (3) rotor, (4) viewing box, (5) cover plate, (6) paraffin wax plug, (7) rotor end-recess, (8) base plate and (9) circular recess for the rheometer Peltier plate. The fluid inlets are slightly offset from the midplane, and are hence not visible in this diagram.

40 toN

R UU

Figure 2.6: The TCL device mounted on the AR-G2 rheometer. The cell gap contains a rheoscopic suspension of mica flakes, and six Taylor vortices are clearly visible at Re = 156.

41 processing conditions. The rotor was also made taller, to the extent possible within size constraints, giving an aspect ratio of ( = 6.0 for the upgraded cell. Although still below the ideal range of ( > 10, the higher aspect ratio further reduces end effects in torque measurements, and ensures that fluid friction on the stem and top flat of the rotor contribute very little to the rotor torque and can be safely ignored. The stator was enlarged to accommodate the bigger rotor, and the radius was chosen to be exactly two inches so that stock-size cast acrylic tubes could be used directly without having to machine the inner bore; boring would remove the pristine as-cast surface, and the tube would require polishing afterwards to restore optical clarity. The TCL device has a gap ratio of c = 0.33, much smaller than TCO and TCS devices; the effects of flow curvature are therefore weaker, inasmuch as TC flow approaches ideal Couette flow between plane walls in the limit of c approaching zero. Furthermore, the radius ratio of r1 = 0.75 is within the range of values for which extensive experimental data on TC flows can be found in the literature; some of these past experimental studies are listed in Table 2.2, and were used as benchmarks to validate the flow characteristics of the TCL device. The stator cylinder and the enclosing viewing box of the TCL device form a single unit, made of optically clear cast acrylic, with all joints bonded permanently by solvent welding. The base plate is made of aluminum, and has a circular recess at the bottom that fits the Peltier plate of the rheome- ter. The acrylic stator unit was fabricated by Boston Plastics Manufacturing Inc. dba PolyFab, Wilmington, MA; the aluminum base plate and rotors were machined at Mach Machine Inc., Hudson, MA. The stator unit was attached securely to the base plate using eight stainless steel (SAE 304) ma- chine screws, and a nitrile rubber gasket (Shore hardness 60A) was placed between the mating faces to ensure a watertight seal. As in the TCS device, two fluid inlets were drilled into the base plate, and were fitted with self clos- ing quick-disconnect sockets at their outer ends. The rotors were machined from aluminum 6061-T6 alloy, with a radial runout less than 0.50 mm (4 % of the cell gap) when mounted on the rheometer spindle. Unlike previous designs, these rotors have a hollow interior to reduce weight and rotational inertia, and are open at the bottom end, to trap air inside the rotor cavity and create a shear-free interface during flow tests. The rotors, as machined, are positively buoyant - they weigh only about 170 g in air but displace over 350 g of water when submerged in the stator. To add more weight and reduce upthrust on the rheometer spindle, the inside cavity was partly filled with

42 Table 2.2: Some previous experimental studies on high Reynolds number TC flows. The radius ratio 71, the aspect ratio ( and the range of operating Re numbers of the TC apparatus are listed; Rej = Ri~id/v and Re, = RQod/v are respectively the inner and outer Reynolds numbers. Where a range for Re, is not mentioned, the TC apparatus had a stationary outer cylinder.

Reference 1 Reynolds number range 0.680 8.5 [73] Wendt (1933) 0.850 18 4.0 x 102 < Re < 1.0 x 105 0.935 42

[70] Smith and Townsend (1982) 0.667 23.7 7.3 x 10 3 < Rei < 1.2 x 105

0 < Rei 2.0 x i0 3 [68] Andereck et al. (1986) 0.883 30 0

[71] Lathrop et al. (1992) 0.725 11.5 8.0 x 102 < Rej < 1.2 x 106

0 < Re2 < 2.0 x 106 [74] van Gils et al. (2011) 0.716 12.3 - -1.4 x 106 < Re, < 1.4 x 106

paraffin wax before the rotors were used in flow tests. For each rotor, roughly 130 g of paraffin wax (Sigma-Aldrich, m.p. > 65 0C as per ASTM D87) was melted and poured inside the rotor cavity and allowed to solidify, increasing the total dry weight to about 300 g. The wax plug can be removed after tests by heating the rotor slightly, and was therefore preferred to polymer-based casting compounds that set permanently in place. For flow tests, the superhydrophobic surface or coating is applied to the outer cylindrical surface of the rotor. The top flat and stem of the rotor are not coated, and remain hydrophilic during the test. In the case of plain, untreated rotors used in baseline measurements, the machined outer surface was further polished to ensure it remains hydrodynamically smooth even at the largest Reynolds numbers; the surface of the rotor was sanded with progressively finer grades of sandpaper - 600, 1200, 1500 and 2000 grit - and then buffed to a mirror finish using a 10 pm fine abrasive paste. After surface treatment, the rotor is attached to the spindle and lowered into the stator, leaving a small 100 pm clearance at the bottom to prevent contact

43 with the base plate during rotation. The stator is then sealed off at the top with an acrylic cover plate, and a neutral cure RTV silicone sealant (Dow Corning 737) is applied to the joint, fixing the lid firmly in place. The silicone sealant hardens over time, and the cover plate has to be removed soon after the test before the sealant cures and bonds the joint permanently; the spent sealant is scraped off and fresh sealant is reapplied to the lid seat before every run. Once sealed, the stator is filled with liquid through one of the two fluid inlets, till the rotor body is submerged and the liquid height rises to a set level about halfway up the stem; care is taken to ensure that a uniform, reflective plastron forms on the superhydrophobic test surface, and to prevent air bubbles and trapped air pockets inside the cell gap. The fluid-filled space above the top flat of the rotor isolates the plastron from ambient air, and provides a robust liquid seal that prevents air ingress into the flow cell. Finally, the outer viewing box is filled with glycerol, and the rotor surface is imaged during the test using a Canon EOS 7D digital SLR camera. Figure 2.6 shows the TCL device, fully assembled and mounted on the rheometer, during a test run.

2.3.3 The AR-G2 rheometer A key challenge in the design of a Taylor-Couette flow facility is the accu- rate measurement and control of rotor speed and torque. To circumvent the complex task of installing and calibrating individual transducers and data acquisition equipment, all three TC devices - TCO, TCS and TCL - were designed as removable fixtures that can be mounted on a readily available AR-G2 rheometer, capable of very precise speed and torque measurements. This section briefly describes some of the main design features of the AR-G2 rheometer; technical specifications for the AR-G2, provided by the instru- ment manufacturer, are listed in Table 2.3. The AR-G2 is a line of commercial CMT 4 rheometers manufactured by TA Instruments, New Castle, DE. It has a fixed bottom plate with a normal force sensor and thermoelectric temperature control, and a rotating spindle at the top to which a standard cone or plate fixture can be attached. The spindle is driven by a low-inertia drag cup induction motor, with the driving torque set by the current input to the motor field windings; the AR-G2 is thus

'Stands for 'combined motor and transducer', a common design for controlled-stress rheometers. By contrast, controlled-strain instruments like the ARES-G2 are of the sepa- rated motor and transducer (SMT) type.

44 Table 2.3: Technical specifications for the AR-G2 rheometer. The listed values are for steady shear operation; specifications differ in the the case of oscillatory tests. The abbreviation 'CR' stands for 'controlled-rate' mode.

Specification Value Unit Maximum torque 2.0 x 10-1 Nm Minimum torque (CR) 1.0 x 10-8 Nm Maximum speed 3.0 x 102 rad s- 1 Minimum speed (CR) 1.4 x 10-9 rad s-1 Angular displacement resolution 2.5 x 10-8 rad a controlled-stress rheometer by design, and uses feedback control to operate in controlled-strain mode. The angular position of the spindle is detected continually by a rotary optical encoder, permitting the angular speed and acceleration of the spindle to be computed accurately. The spindle, along with the motor assembly, bearings and the optical encoder, are mounted on a movable head which can be positioned at the required working height, with micrometer accuracy, by a precision linear ball screw drive. Inside the head, the spindle is supported entirely by magnetic and pneumatic bearings; there is no physical contact between fixed and moving parts, resulting in near-zero friction on the spindle during rotation. Axial support is provided by a magnetic bearing, consisting of an iron thrust plate fixed to the the spindle and held suspended by two electromagnetic actuators placed above and below the plate; the vertical position of the thrust plate is monitored and rigidly maintained by a closed-loop controller. Lateral stiffness is provided by two porous carbon radial air bearings located near the top and bottom ends of the spindle; during operation, these bearings require compressed air supply at 30 psi line pressure. At large driving torques, the motor temperature rises and active cooling is required to counter thermal expansion of the drag cup. Motor cooling with compressed air starts automatically when the torque exceeds a preset value of about 35 mN m. At high angular speeds, the additional friction from the cooling airflow produces a systematic error in torque measurements; therefore, to ensure accuracy of experimental data, flow tests using the TCL device were restricted to torques below 35 mN m (corresponding to a wall shear stress of T = 50 Pa), and data points obtained after the onset of motor cooling were discarded. This limits the maximum angular speed attainable

45 with the TCL device to about 160 rad/s for most test surfaces, whereas TCO and TCS devices can reach angular speeds up to 250 rad/s without exceeding the threshold torque for cooling.

2.4 Flow characterization

Rayleigh (1916) was the first to propose a mathematical criterion for the centrifugal stability of swirling flows; he reasoned, by analogy with the case of a stratified fluid column under gravity, that any two-dimensional swirling flow 5 of an inviscid fluid would be stable to axisymmetric disturbances if the square of the circulation increases monotonically with radius 165]:

d (rVo)2 > 0 (2.8) dr Synge (1933) derived this inequality rigorously by linear stability analysis, and showed that it was both a necessary and sufficient condition for stabil- ity, in the case of an inviscid fluid and for axisymmetric disturbances 175]. If the fluid is instead viscous, he found that this condition was still suffi- cient, although not necessary, to ensure stability of the flow to infinitesimal, axisymmetric perturbations [76]. The inequality (2.8) is known as the Rayleigh criterion, and we may use it to examine the stability of the primary flow inside the TC apparatus. Consider the steady shear flow of a viscous fluid between two concentric cylinders of radii RI and R0, rotating with angular velocities Qj and Q,. The simplified momentum equation in the azimuthal direction reads I d ( dVo Vo rr=-- 0 (2.9) ( r dr dr r2 and it admits a solution of the form

Vo(r) = Ar + -, (2.10) r with the constants A and B set by the no-slip boundary conditions at the two cylindrical walls:

Vo(Ri) = Rjnj, V(Ro) = Ro0 o 'We use the term 'swirling flow' to denote flows with V, = 0, V = 0, and the azimuthal velocity Vo(r) a function only of the radial coordinate r, in cylindrical coordinates.

46 Solving for A and B, we get

A - R - - R Ri (B - (2.11) R2 R2 -- R?

We next determine the conditions under which the velocity profile (2.10) will satisfy the Rayleigh criterion. Substituting equation (2.10) into (2.8) yields

rA(Ar 2 + B) > 0 (2.12) which reduces to the condition

(QRO - QjRi) Vo(r) > 0 for R < r < RO (2.13)

This inequality is true only if the two cylinders rotate in the same direction, and the circulation at the outer cylinder is larger than that at the inner cylinder; if both conditions are met, the primary shear flow inside the TC apparatus remains stable. In particular, the Rayleigh criterion is always satisfied for a fixed inner cylinder and a rotating outer cylinder; no secondary vortical flows develop in this case, and the initial Couette flow transitions directly to turbulent shear flow as the speed of rotation is increased. By contrast, the flow between a rotating inner cylinder and a stationary outer cylinder is Rayleigh unstable, and secondary flow patterns appear as the speed of rotation is increased. The critical rotation rate at which the pri- mary Couette flow becomes unstable to axisymmetric perturbations was first derived by Taylor (1923). Above this critical speed, linear stability theory6 predicts the onset of a steady and axisymmetric secondary flow, consisting of axially periodic pairs of counter-rotating toroidal vortices, known as Tay- lor vortices [66]. If the speed is increased above a second critical value, the Taylor vortex flow (TVF) itself develops a secondary instability in the form of azimuthal traveling waves, leading to wavy vortex flow (WVF) [67]. At even larger speeds, the waves disappear and turbulent Taylor vortices (TTV) are observed. Further increase in speed causes the vortex structure to grad- ually break up, and the flow transitions to full turbulence (TUR) with no discernible large-scale features 168,71]. If both inner and outer cylinders can rotate, several other flow states in addition to those mentioned here are possible, depending on the speed and 6 1n the context of weakly nonlinear theory, this primary instability corresponds math- ematically to a steady supercritical pitchfork bifurcation.

47 direction of rotation of the two cylinders; these flow states are not always unique, and more than one may be stable under the same flow conditions. Andereck, Liu and Swinney (1985) identified as many as 18 distinct flow regimes in their experimental TC apparatus, spanning a complex and diverse transition diagram [68]. Although the TCL device was finally intended to op- erate exclusively in the turbulent regime, it was interesting to see which other flow states and transitions occur en route to turbulence; these low Reynolds number flow states were observed using flow visualization, as outlined in the next section.

2.4.1 Flow visualization Flow patterns inside the TCL device were visualized with the help of a rheo- scopic fluid, prepared by mixing 4g/L of finely powdered synthetic mica 7 in a glycerol-water mixture of suitable viscosity; an 80 % glycerol solution was used for low Reynolds numbers upto 500, a 50 % solution was used for intermediate Reynolds numbers between 500 and 10 000, and pure deionized water was used to observe the flow at Reynolds numbers above 10000. In each case, the cell gap was filled with the mica suspension, and the flow was imaged as the rotor accelerated gradually from rest to its maximum speed; the angular acceleration of the rotor was 0.25rad/s 2 or lower in all experi- ments. The flat, shiny mica crystals, having sizes between 15 pm and 150 pm, reflect the ambient light and align in the local direction of flow; as a result, regions of predominantly radial secondary flow appear dark, compared to re- gions of axial flow where mica flakes are oriented normal to the line of sight of the camera. For instance, in TVF, the vortex rolls appear bright, whereas the boundary between adjacent vortices appear dark; the inflow boundaries usually appear darker and more well defined than the outflow boundaries. To further enhance visual contrast during imaging, the solution itself was dyed green using food coloring, to provide a dark background for the white mica particles. Representative images of flow patterns observed in the TCL device at different Reynolds numbers are shown in figures 2.7 and 2.8; in estimating the value of Re, changes in fluid viscosity caused by the presence of mica particles were ignored. The flow states and transitions observed in the TCL device were largely consistent with previous observations on TC systems with a rotating inner

7 Approximately 89 % synthetic fluorphlogopite and 11 % anatasic titanium dioxide.

48 (a) Re = 20 (b) Re = 400

(c) Re = 500 (d) Re = 700

Figure 2.7: Flow patterns observed in the TCL device at different Reynolds num- bers during flow visualization experiments. (a) laminar azimuthal Couette flow (AZI) at Re = 20; (b) Taylor vortex flow (TVF) with 10 vortices at Re = 400; (c) Taylor vortex flow with 8 vortices at Re = 500; (d) Taylor vortex flow with 6 vortices at Re = 700.

49 (e) Re = 1500 (f) Re = 8000

(g) Re = 80000 (h) Ekman cells

Figure 2.8: (Continued) Flow patterns observed in the TCL device at different Reynolds numbers during flow visualization experiments. (e) wavy outflow bound- aries (WOB) at Re = 1500; (f) turbulent Taylor vortices (TTV) at Re = 8000; (g) featureless turbulence (TUR) at Re = 80000; (h) pair of Ekman cells become visible after the flow is suddenly stopped from a high Reynolds number.

50 cylinder and a stationary outer cylinder 168]. At very low Reynolds numbers, laminar azimuthal Couette flow is observed, with shear boundary layers at the top and bottom end walls. As the speed of rotation is increased, Taylor vortices appear, possibly triggered prematurely by B6dewadt layers which develop at the top and bottom end walls. The circulation induced by the end wall flows appear as a pair of weak Ekman cells each extending nearly half the height of the cell; the presence of these cells becomes particularly evident after sudden cessation of high speed flow, as seen in Figure 2.8(h).

The TVF state most often observed in the TCL device was a stable six- vortex state, although, depending on the initial rate of acceleration, less sta- ble eight-vortex and ten-vortex states were also sometimes produced. The ten-vortex TVF state quickly became unstable as the speed was increased, and devolved into the eight-vortex state by the merger of adjacent pairs of vortices. The eight-vortex TVF could be sustained to a higher Reynolds number, provided the rate of acceleration was kept sufficiently low; at higher speeds, it transitioned to an eight-vortex WVF, before becoming unstable, losing a pair of vortices, and returning to the stable six-vortex state. Inter- estingly, the six-vortex state did not exhibit a wavy vortex flow regime as the speed was increased, possibly due to the low aspect ratio of the TCL ge- ometry; instead it passed through states resembling wavy outflow boundary (WOB) and wavelet (WVL) flows, before finally transitioning to turbulent vortex flow (TTV). With further increase in Reynolds number, the TTV structure was found to gradually dissipate until the flow in the gap became uniformly turbulent. Beyond the critical Reynolds number for transition to turbulence, calculated from torque measurements, the original six-vortex structure was no longer discernible; however, circulation due the end wall boundary layers as well as vortex-like coherent structures could still be per- ceived in the flow even at the highest Reynolds numbers, an observation that is consistent with previous experimental studies [77].

The purpose of flow characterization was to determine the critical Reynolds number at which flow inside the TCL geometry becomes fully turbulent, a necessary step before the device can be used for friction measurements on test surfaces. The simple flow visualization technique used here was of insufficient resolution to locate this transition point accurately. The transition Reynolds number Ret was instead determined by a more reliable method based on torque measurements, which is described in detail in the next section.

51 Table 2.4: Density and viscosity of working fluids used in flow tests in the TCL device. The properties of water and synthetic seawater were taken from standard reference tables [78,79], and of the two glycerol solutions were determined experi- mentally. All values correspond to a temperature of 25 C.

Liquid Density Dynamic viscosity Kinematic viscosity kg m- 3 Pas m 2 sI Deionized water 997 8.90 x 10-4 8.93 x 10-7 Synthetic seawater 1024 9.59 x 10-4 9.37 x 10-7 57% glycerol in water 1107 1.30 x 10-2 1.17 x 10-' 84 % glycerol in water 1218 1.18 x 10-1 9.70 x 10-'

2.4.2 Transition to turbulent flow

Baseline flow curves for the TCL device were obtained by measuring the an- gular speed as a function of torque, on a smooth aluminum rotor. Deionized water and two different glycerol-water mixtures were used to obtain high resolution friction data over five decades of flow Reynolds numbers, in the range 4.0 x 10-1 < Re < 8.6 x 104. The density and viscosity of the work- ing fluids are listed in Table 2.4; these values were obtained from property tables [78,791 in the case of water, and determined experimentally for the two glycerol solutions. Densities were measured by weighing exactly 10 mL of each solution with a microbalance. The viscosity of both solutions were measured on the AR-G2 rheometer using a 60 mm 2' cone and plate geom- etry, and found to be constant, within experimental error, for shear rates between 100 s-1 and 600 s-. All fluid property measurements and flow tests were performed at a temperature of (25 2) C. To improve accuracy of measurements, the baseline tests were performed with the AR-G2 operating in the controlled-stress mode; the torque T was increased in steps from 100 pN m to 35 481 pN m, and the angular speed Q of the rotor was measured. With water as the working fluid, an approximate time scale for viscous diffusion of momentum across the cell gap is given by At = d2 /v = 181s. At each torque step, therefore, speed measurements were started after a 3 min waiting time, to ensure that the flow had reached steady state. This was deemed sufficient, since the actual time required to reach steady state is expected to be considerably smaller than At, because of convection from secondary flows, and turbulent diffusion at high Reynolds

52 numbers. For the two glycerol solutions having much larger viscosities than water, a shorter waiting time of 1 min was used. After this waiting period had elapsed, the angular speed of the rotor was recorded at Is intervals over the next 30 s, and the 30 raw data points were averaged to give the final value of Q. To compare data obtained using liquids of different viscosities, it is useful to define a non-dimensional torque G

G = (2.14) and plot this against the flow Reynolds number Re, as shown in Figure 2.9. In these coordinates, all three data sets collapse into a single characteristic curve for the TCL device; furthermore, the data from all three tests agree remarkably well in their regions of overlap. To determine the transition Reynolds number Ret from this friction curve, we adopt the same method used by Lathrop, Fineberg and Swinney (1992). At large Reynolds numbers, we assume a power law scaling for torque, of the form

G ~ Re' (2.15) and compute the torque exponent a:

d log G dlog=d log Re e(2.16) The data set for water consists of discrete data points spaced at uniform log intervals of A (logo G) = 0.05, and spans the range of Reynolds numbers from 2.1 x 103 to 8.6 x 104. Since direct numerical differentiation accentuates noise in the data, the value of a was instead calculated using a sliding linear least squares fit over 15 adjacent data points (equivalent to a log interval of A (log1 o G) = 0.70). The exponent a is plotted as a function of Reynolds number in Figure 2.10; it increases monotonically with Re, showing that there is no single power law scaling applicable in this range of Reynolds numbers. However, a sharp discontinuity in slope is observed near Re = 10 975, indicating a change in the dependence of a, and by extension G, on the Reynolds number Re. Lathrop et al. identify this as the point of transition to shear-driven turbulence, beyond which the flow in the cell gap behaves similar to wall-bounded shear flows with a turbulent boundary layer

53 109 --

108

107

2 106

C

105

103_

102

100 101 102 1i104 105 Flow Reynolds number Re

Figure 2.9: Baseline friction curve for the TCL device. Data was obtained using water ( * ), 57% glycerol solution ( 9 ) and 84 % glycerol solution ( * ) as the working fluids, for different Re ranges.

54 1.7- Ret = 10975 / 1.6 -

W 1.5 -

H 1.4-

1.3 -, , 103 104 105 Reynolds number Re

Figure 2.10: The local torque exponent a as a function of the flow Reynolds number Re. A change in slope occurs at Ret = 10975, indicating transition to shear-driven turbulent flow. The red line is a piecewise linear fit for a near the transition point, given by equation (2.18).

[711. The value of the transition Reynolds number for the TCL device is therefore Ret = 10 975 (2.17) An approximate, empirical relation for a can be obtained by separate linear fits on the data before and after the transition point:

a 1.59 + 0.469 log1 o (Re/Ret) for Re < Ret (2.18) 1.59 +0.172loglo(Re/Ret) for Re> Ret The value of a at the transition point is about 1.59, which is close (but not exactly equal) to the marginal stability exponent of 5/3 171]. The value of Ret for the TCL device, with a radius ratio 7 = 0.75, com- pares well to the value of 1.3 x 104 obtained by Lathrop et al. for their TC apparatus, which had a radius ratio 17= 0.72. All measurements on test sur- faces in the TCL device were performed at Reynolds numbers Re > 15000, well above the transition Reynolds number; this was done to ensure that the flow inside the device had fully transitioned to turbulence, even in the presence of wall slip produced by the drag reducing surface being tested.

55 2.4.3 Torque scaling in turbulent TC flow In the fully turbulent regime, a scaling law for friction on the inner cylinder can be derived by asymptotic matching of either the velocity or the angular momentum, in the region of overlap of the core flow and the two wall layers 171,80]. The shear stress on the inner wall at steady state is given by

T = (Tr) = pR d (2.19) dr _ r=R Here, y is the dynamic viscosity of the fluid, and the notation (---) is used to denote time averaged quantities. For simplicity, we neglect all end effects, and assume that the shear stress is uniform over the lateral surface of the rotor. The steady frictional torque on the rotor is therefore

T T .21rRiL -Ri (2.20)

Rearranging, we get an equation to compute r from the measured torque: T T = T 2(2.21) 27Ri L A useful non-dimensional form of the wall shear stress is the friction factor Cf, defined as 2T Cf pV2 (2.22) Near the inner and outer cylinders, the usual wall scaling for turbulence applies; the turbulent velocity fluctuations and the Reynolds stresses scale with the friction velocity u*, given by

U* = (2.23)

In the core flow, momentum is transported principally by Reynolds stresses, and the relevant length and velocity scales for the turbulent motion of the fluid are, respectively, the gap size d and the friction velocity u*. We can therefore define a shear Reynolds number Re*, analogous to the flow Reynolds number Re, using the length and velocity scales in the core region:

Re* = u (2.24)

56 On the other hand, all Reynolds stresses must vanish close to the wall, and momentum transport in the wall layers is dominated by viscous stresses. The length scale in the near-wall layer is thus set by the shear stress at the wall and the viscosity of the fluid; this viscous length scale is given by

pv2 V 6V _ (2.25) T U*~

Notice that Re* is simply equal to d/6, or the width of the cell gap expressed in wall units. The following relationship between the various quantities can be easily derived by rearranging equations (2.22), (2.23) and (2.24):

Re Vi 2 .2 6 ) RRe* * u*- - Cf- (2

Panton (1992) used matched asymptotic expansions for the angular mo- mentum r (Vo) of the fluid, to derive a logarithmic scaling for friction in high Reynolds number turbulent flow between a rotating inner cylinder and a sta- tionary outer cylinder [80]. The flow is assumed to consist of an inviscid core region of nearly constant angular momentum, flanked by thin wall lay- ers at the inner and outer cylinders; this is consistent with the observations of Smith and Townsend (1982), who used hot-wire anemometry to measure velocity profiles of the turbulent flow inside an experimental Taylor-Couette apparatus [70]. Starting from the Reynolds-averaged Navier-Stokes equa- tions, and using general scaling arguments, Panton obtained a skin friction scaling law of the form

= M log Re*+N (2.27)

Here M and N are constants that depend only on the radius ratio 'q of the two-cylinder system, and the von Kdrmdn constant K; in limit of zero cur- vature, the value of M is expected to approach 2/K, which is approximately 4.88 for K = 0.41. It is interesting to note that the friction law for Taylor- Couette turbulence has the same logarithmic functional form as the friction law for other kinds of wall-bounded turbulent flows, such as pipe flow and boundary layer flow, even though the dynamics of turbulence in these flow systems are very different.

57 28- - Cq27-

26 - ~25 -- 9' 4-D 0/I ~24--

23

cc 22 21 I I I I 700 1000 1400 2000 3000 Shear Reynolds number Re*

Figure 2.11: Skin friction curve for a smooth rotor in the TCL device, plotted in Prandtl-von Karmin coordinates. Filled circles ( 0 ) represent data obtained using water. The dashed line is a linear least-squares fit to this data, giving M = 4.52 and N = -8.29 in equation (2.27). Empty circles ( o ) represent data obtained using ASTM D1141 synthetic seawater.

For the TCL device, conformance with the logarithmic scaling law (2.27) for Re > Ret was verified using a semi-log plot of \ 2 /Cf against the shear Reynolds number Re*, also known as a Prandtl-von Kdrmdn plot. In this case, the baseline torque data was obtained using a smooth untreated alu- minum rotor with deionized water (18.2 MQ cm resistivity) as the working fluid. All measurements were made at a temperature of (25 2) 'C. The same test protocol used in flow tests on drag-reducing surfaces was followed; the steady state torque on the rotor was measured at ten different angular speeds Q ranging from 30.20 rad s-1 to 158.5 rad s-1, which corresponds to Reynolds numbers between 1.64 x 104 and 8.59 x 104. At each speed, torque measurements were started after a 15s waiting period to ensure that the flow had reached steady state. The rotor torque was sampled at 1s intervals

58 over 30 s, and the 30 raw data points were averaged to give the final value of torque; the relative standard deviation of the torque values was found to be less than 1 % at all speeds. Since the experiment was performed with the rheometer operating in the controlled-strain mode, the consistency of data was checked against the speed-torque curve obtained previously using a controlled-stress protocol; the two data sets were concordant within the limits of experimental accuracy. Figure 2.11 shows the friction curve for the TCL device in Prandtl- von Karmin coordinates. The data points fit very well to a straight line, in accordance with equation (2.27). The value of the constants M and N for the TCL device were obtained by linear least-squares regression on the test data for water: M = 4.52, N = -8.29 (2.28) The characteristic flow curve for the TCL device in the fully turbulent regime is therefore

2 _ - 8.29 (2.29) Cf=Cff 4.52 log Re* A similar test was performed using synthetic seawater 8 purchased from Ricca Chemical Company, Arlington, TX; the density and viscosity of this synthetic seawater solution were obtained from property tables, and are listed in Table 2.4. The friction data from this test is also plotted in Figure 2.11, and shows excellent agreement with the baseline curve obtained using water. The friction law (2.27) can be used to obtain an expression for the torque exponent a. Using equations (2.26) and (2.27), we get Re = Re* (M log Re* + N) (2.30) This equation can be inverted to give Re* as a function of Re: Re' Re* = R'(2.31) W (eN/MRe') Here Re' = Re/M, and W(x) is the Lambert W function.9 From equations

8Substitute ocean water without heavy metals, prepared as per ASTM D1141. The salinity of this solution is approximately 35 g/kg. 9The Lambert W function W(x), also known as the product logarithm function, is defined as the functional inverse of f(x) = xex, and satisfies the functional equation x = W (xex). For real x > 0, the function W(x) is single-valued and non-negative. Also W(O) = 0, and W(x) increases monotonically with x.

59 1.8

S 0 0 0 1.6 (t 0 0 S 0 0 0 0 S S 0 0 -0 1.4 S

1.2 - - - 1 2 3 4 5 6 Reynolds number Re .104

Figure 2.12: The local torque exponent a as a function of the flow Reynolds number Re. The solid red line is the theoretical prediction for a, equation (2.33), with M = 4.52 and N = -8.29.

(2.14), (2.24) and (2.31), we have

27 f2 27rR e 2 G = (Re*) = 2irRi Re' (2.32) d2 d2 W (eN/MRe')I

We can now derive an explicit expression for the torque exponent a:

dlnG 2 a 2 - (2.33) d In Re 1 + W (eN/MRe')

As Re goes to infinity, the expression on the right hand side asymptotes to the value 2. This behavior is consistent with the fact that the torque exponent a cannot exceed 2, a strict upper bound which can be rigorously derived from the Navier-Stokes equations 181]. Figure 2.12 compares equation (2.33) with the value of a calculated using a sliding linear fit on experimental data; there is fair agreement at Reynolds numbers just above Ret, and an excellent fit is obtained for Re > 4 x 10'.

60 2.5 Closure

This chapter describes the design and characterization of an experimental Taylor-Couette facility for measuring skin friction on drag reducing surfaces under turbulent flow conditions. The original TCO apparatus designed by Srinivasan et al. [60] provided the basis for building an upgraded TC system, namely the TCL device, which was subsequently used for all flow tests re- ported in this study. The flow regimes in the newly built device were studied using flow visualization, and the point of transition to turbulent flow was accurately located by means of baseline friction measurements on a smooth, uncoated rotor. Beyond this transition point, the friction curve for the TCL device follows a logarithmic scaling law, consistent with previous theoretical and expcrimcntal studies in the literature [70,71,80]. The details of the ac- tual flow tests performed on various drag reducing surfaces using the TCL device are discussed fully in the next chapter.

61 62 Chapter 3

Characterization of Drag-Reducing Surfaces

3.1 Introduction

For practical drag reduction applications, we require scalable superhydropho- bic surfaces that can be easily applied over large areas of the solid boundary in contact with the flow [1,7]; for example, the wet hull area of ships and underwater vehicles extend over several square meters. Regularly patterned superhydrophobic surfaces, consisting of periodic arrays of ridges or posts, have been extensively investigated for anti-wetting and drag reduction ap- plications over the past two decades; accurate scaling laws governing slip length over these 'canonical' surfaces have been derived by theoretical and computational studies [51,55]. However, fabrication of these surfaces require expensive and precise manufacturing techniques, which limits their scalabil- ity to real-life applications. Randomly rough superhydrophobic surfaces, on the other hand, can be produced by relatively inexpensive and easily scalable techniques, like spray- coating, sandblasting, and chemical or electrochemical etching, that are al- ready in widespread use in industrial surface treatment processes. However, the morphology of these surfaces is more difficult to characterize, and a scal- ing law for slip length in terms of surface roughness parameters is yet to be derived. Although rough superhydrophobic surfaces have been successfully applied in recent years for non-wetting applications, including several com- mercially available anti-rain and water-repellent surface treatments [82,831,

63 they have not been systematically characterized for drag reduction applica- tions, particularly under turbulent flow conditions. To successfully design rough superhydrophobic surfaces for deployment in real-life marine and undersea applications, we need a quantitative under- standing of the relationship between surface morphology and drag reduction performance, so that large-scale surface treatment processes can be carefully tuned to generate the optimal surface texture. Whereas a large number of randomly rough surfaces have shown extraordinary success in anti-wetting applications, most of them do not perform equally well in drag reduction; indeed, many excellently superhydrophobic surfaces are found to produce no drag reduction in turbulent flow, even when they remain perfectly non-wet underwater [7]. This underlines the necessity to investigate in detail the par- ticular features of the surface texture which confer the ability to reduce drag under turbulent flow conditions. In this study, we focus on characterizing the drag reduction performance of rough superhydrophobic surfaces, and attempt to correlate it to statistical measures of the random surface texture. The TCL apparatus, described in detail in the previous chapter, was used to perform skin friction measure- ments on five model drag reducing surfaces, of which four were provided by collaborating research groups in the MURI project; the details of the test surfaces and the collaborating teams are listed in Table 3.1. Three of the surfaces investigated were randomly rough superhydrophobic surfaces, pro- duced by various easily scalable mechanical and chemical texturing processes. The other two consisted of periodic streamwise grooves, containing air or a low viscosity liquid lubricant; the groove pattern can be produced on metal- lic substrates by conventional machining techniques, or using laser beam machining or die-sink electrical discharge machining. The drag reduction performance of these surfaces were assessed under moderately turbulent flow (Re < 10), and the effective slip length on each surface was estimated fol- lowing the work of Srinivasan et al. (2015) [60]. Additionally, height profiles of the rough superhydrophobic surfaces were measured by non-contact opti- cal profilometry, and the size, distribution and spacing of roughness elements were quantified by computing a number of surface statistical parameters. Flow tests, in conjunction with surface profile measurements, allow us to draw correlations between specific features of the surface texture and par- ticular traits desirable in turbulent drag reduction applications - mechanical durability, large slip lengths and excellent resistance to pressure induced wet- ting. The subsequent sections of this chapter describe the fabrication and

64 Table 3.1: List of drag-reducing surfaces used in flow tests and surface characterization studies. Four of the five test surfaces were provided by collaborating research groups from other universities as part of the ONR MURI project.

Test surface Collaborators Affiliation References

Acrylic FPOSS A. Rajappan Dept. of Mechanical Engineering [36] Srinivasan et al. (2011) spray coating G.H. McKinley MIT, Cambridge, MA [60] Srinivasan et al. (2015)

Etched Al K.B. Golovina aDept. of Materials Science and Engineering 138] Yang et al. (2011) bDept. M. Bobanb of Macromolecular Science and Engineering [841 Ling et al. (2016) B. Tobelmanna University of Michigan, Ann Arbor, MI A. Tuteaa,b c-,1 Sandblasted, V. Pillutla Dept. of Mechanical Engineering V. Pillutla etched and Abhijeet University of Texas at Dallas, Richardson, TX (private communication) boehmitized Al W. Choi

Liquid-infused T. Van Buren Dept. of Mechanical and Aerospace Engineering [85] Rosenberg et al. (2016) grooves A.J. Smits Princeton University, Princeton, NJ [86] Van Buren et al. (2017)

Air-filled J. Wang Dept. of Mechanical and Nuclear Engineering [39] Kim et al. (2013) grooves X. Dai and Materials Research Institute [87] Yang et al. (2014) T.-S. Wong Pennsylvania State University, University Park, PA [88] Dai et al. (2015) [89] Wang et al. (2016) experimental characterization of prototype drag reducing surfaces investi- gated as part of this study.

3.2 Experimental methods

Rough superhydrophobic surfaces were characterized experimentally by a combination of flow tests, surface profile measurements and contact angle goniometry. Skin friction curves were first obtained for all test surfaces in shear-driven turbulent flow, for Reynolds numbers between 1.64 x 104 and 8.59 x 104, and at shear stresses ranging from 2 Pa to 50 Pa; from this data, a characteristic slip length was determined for each surface. For each of the three rough superhydrophobic surfaces, height profiles were measured using a laser profilometer, and used to calculate surface statistical parameters that quantify lateral and vertical length scales of roughness. Finally, the chemical hydrophobicity of the surfaces was assessed through contact angle goniometry. The details of experimental procedure and data analysis used in these characterization tests are described in the next three sections.

3.2.1 Skin friction measurements Skin friction measurements were performed on all test surfaces in the TCL device, using water (deionized to 18.2 MQ cm resistivity) and synthetic sea- water (prepared as per ASTM D1141) as the working fluids; the physical properties of both liquids are listed in Table 2.4. All flow tests were per- formed at a temperature of (25 2) 'C. At the start of each test, the sub- merged rotor was inspected to ensure that a uniform, reflective plastron was present in the case of superhydrophobic surfaces, and that all grooves were fully filled with air or lubricant in the case of streamwise grooved surfaces. In the TCL device, the plastron on the test surface is completely isolated from outside air by the fluid-filled space above the rotor. To obtain reliable slip length measurements, gas loss from the plastron by diffusion has to be minimized, for which the concentration of dissolved air in the working fluid must be close to its saturation value at the test temperature. Therefore, before each test, the required volume of working fluid was allowed to aerate naturally inside a ventilated, partly filled container at 25 C for 12-18 h, so that diffusional equilibrium is attained by the exchange of gases across the free surface of the liquid; forced aeration was not used due to the risk of

66 Table 3.2: The angular speeds and the corresponding flow Reynolds numbers at which frictional torque on the rotor was measured during flow experiments in the TCL device.

Step Rotor speed Q Reynolds number Re rad s-1 Deionized water Synthetic seawater 1 30.20 16370 15597 2 36.31 19682 18753 3 43.65 23660 22544 4 52.48 28446 27104 5 63.10 34203 32589 6 75.86 41119 39179 7 91.20 49434 47102 8 109.6 59408 56604 9 131.8 71441 68070 10 158.5 85914

accidental oversaturation. The same test procedure was followed in experiments on all drag-reducing surfaces; the angular speed Q of the rotor was increased in ten steps from 30.20 rad s- 1 to 158.5 rad s-1, and the steady torque T on the rotor at each speed was recorded. The rotor speeds and the corresponding Reynolds num- bers for the ten flow steps are listed in Table 3.2. The lowest angular speed corresponds to Re = 16370 in water and Re = 15597 in seawater, and in both cases is well above the transition Reynolds number Ret of the TCL device [equation (2.17)]. At each step, an initial 15s waiting period was pro- vided to ensure that the flow had reached steady state; the rotor torque was then sampled at Is intervals for 30s, and averaged to give the final value of torque. In every flow test, the relative standard deviation of the 30 raw torque values was found to be less than 1 % at all speeds, confirming that the flow was indeed at steady state. Between adjacent steps, the rotor speed was increased smoothly from one value to the next by a gradual linear ramp over 30 s, to avoid sudden disturbances that may cause premature failure of the plastron on the test surface. When designing the experiment protocol, the durations of individual flow and ramp steps were kept as short as possible to limit shear damage to the surface, so that friction measurements at the higher rotor speeds are not inordinately affected by surface wear incurred

67 in previous steps. The state of the plastron on the rotor surface, and the drainage of lubricant from the rotor grooves, were continuously monitored during the test, and recorded through the viewing box using a Canon EOS 7D camera. The data from all flow experiments are included in appendix B. The Reynolds number Re and the wall shear stress T were calculated from the rotor speed and torque data, using equations (2.4) and (2.21):

_ Rid T Re = r = (3.1) V 27rRi L

All end effects in torque measurements are ignored in the calculation of T. The percentage decrease in wall shear stress produced by the test surface, which we henceforth refer to as the 'percentage drag reduction' or DR, is given by DR = X 100%, (3.2) TO where O is the baseline shear stress measured on a smooth untreated rotor at the same Re number (or equivalently, the same rotor speed), and using the same working fluid. In all experiments on drag-reducing surfaces, the percentage drag reduction was observed to increase with Re number, reach a maximum, and subsequently decrease as failure set in and the plastron (or lubricant) was gradually sheared away by the flow. A drag-reducing superhydrophobic surface retains a layer of air (plastron) between texture asperities when submerged in water; flow over such a surface therefore encounters intermittent regions of direct solid-liquid contact with the substrate, interspersed between large areas of liquid-air contact where the flow is separated from the solid surface by the trapped air layer [Figure 1.5]. Whereas the usual no-slip condition applies in regions of liquid-solid contact, the liquid-air interface provides an almost shear free boundary over which the flow can slip; the overall effect of the presence of these slip regions is a net reduction in the average shear stress exerted by the flow on the wall. We assume that the inhomogeneities in local wall slip average out in the viscous, near-wall region, so that at a sufficient distance from the wall, the effect of the composite boundary on the outer flow can be modeled as arising from a uniform tangential slip velocity over the whole surface. The time-averaged slip velocity (V,) is related to the wall shear stress r by the Navier boundary condition: (V) = b -(7 b (u)(3.3) P 6V

68 Here, b is a finite, effective slip length on the superhydrophobic surface in wall bounded turbulent flow. Non-dimensionalizing this equation using the inner length and velocity scales in turbulence, we get 160]

(V) + = b+ (3.4) where (V,)+ = (V) /u* is the non-dimensional slip velocity, and b+ = b/6, is the slip length scaled in wall units. With no-slip on the inner cylinder, as in the case of a smooth untreated rotor, the frictional torque in fully turbulent flow in the TCL device scales according to equation (2.27):

= M log Re* + N (3.5) Cff Using equation (2.26), this can be rewritten as

= M log Re* + N (3.6) U* The scaling analysis and asymptotic matching of angular momentum lead- ing to equation (3.5) is unchanged in the presence of a uniform azimuthal slip velocity on the inner wall; the actual speed of the rotor surface Vi can therefore be simply replaced by the apparent or 'effective' speed Vi - (V), to obtain the modified friction law in the presence of wall slip:

K- (VS)= M log Re* + N (3.7) U* Rearranging and using equation (3.4), this becomes

= MlogRe* + N + b+ U* (3.8) This form of the friction scaling law, in the case of a uniform wall slip on the inner cylinder, was derived rigorously by Srinivasan et al. (2015) and experimentally verified in the TCO device on an acrylic fluorodecyl-POSS spray-coated surface [60]. As a further simplification, the dimensional slip length b is assumed constant for a given superhydrophobic surface, indepen- dent of the shear Reynolds number Re*; with this assumption, equation (3.8) can be written as:

=Mlog Re* + N + Re* (3.9) Cff d

69 since b+ = b/6, = (b/d)Re*. At large Re*, turbulent pressure and veloc- ity fluctuations intensify and may cause the liquid-air interface to penetrate deeper into the texture, increasing the wetted solid fraction and decreasing the effective slip length on the surface; there is no reason, a priori, to expect the slip length b to remain constant under changing flow conditions. Exper- imental data, however, is found to fit the predicted curve for a constant b reasonably well, until the point where surface failure commences either by plastron loss or drainage of lubricant; this also corresponds roughly to the point of maximum DR. The constants M and N for the TCL device are known from baseline measurements on a smooth rotor, and are given by equation (2.28):

2 _b = 4.52 log Re* - 8.29 + - Re* (3.10) Cf d

The effective slip length b for each of the five test surfaces was calculated by fitting equation (3.10) to the friction data before failure, in Prandtl- von KArmin coordinates. The divergence of experimental data from the theoretical curve indicates the onset of surface failure, and the value of Re* and T at which this occurs can be used as a measure of the resistance to failure under turbulent flow conditions; we refer to it hence as the 'failure shear Reynolds number' and the 'failure shear stress', respectively.

3.2.2 Surface profilometry The determination of roughness parameters requires accurate measurement of surface heights. Sample height profiles for all three rough superhydropho- bic surfaces were measured using a Keyence VK-X250 laser scanning confocal microscope (Keyence Coroporation, Itasca, IL). A 50X objective lens with a numerical aperture of 0.95 was used in all measurements, giving a lateral resolution of 280 nm and a vertical (height) resolution of 100 nm. To ob- tain the height profile, a 408 nm visible laser beam (spot radius 0.2 lm) is scanned across the surface, and the reflected light intensity is recorded at several successive focal planes at increasing heights; a pinhole confocal opti- cal system is used to block out stray and out-of-focus background light from the PMT sensor, enhancing the lateral resolution of measurements. At each (x, y) coordinate, the location of the surface is then identified at the focal plane corresponding to the peak reflected intensity. The final height data,

70 acquired from a total scan area of 0.062 mm2, is rasterized to a 2D rectangu- lar grid of 1024 by 768 points. After minor filtering to remove speckling and other noise artifacts, and corrections for surface tilt, the raw height data was exported to a text file as a set of discrete (x, y, z) coordinates. All further data processing and calculation of surface parameters were performed using custom MATLAB1 scripts. Before computations, the mean height was first subtracted out so that the profile is centered around the z = 0 plane:

h(x, y)= z(x, y) - z(x, y))A (3.11)

The notation .A. ) here denotes spatial averaging over the sample area A. The most commonly used surface parameter is the root mean square rough- ness w, also called the interface width, which is defined as

w = (h2 (x, y))A (312)

The root mean square roughness quantifies the length scale of height fluctua- tions on the surface. For discrete height data, the root mean square roughness is calculated simply as the standard deviation of surface heights:

w = std2(h) ;

We can similarly compute the root mean square surface slope, defined as

s =(|Vh(x, Y)|) = (h2 +2)A, (3.13) where Vh is the gradient vector on the surface, and hx and hy are the par- tial derivatives of the surface height h(x, y) along the x and y directions. In MATLAB, these partial derivatives were computed using the gradient function, which returns a finite difference approximation for Vh(x, y):

[Gx, Gy] = gradient(h, c); s = sqrt(mean2(Gx.^2 + Gy.^2));

Here c Ax = Ay is grid spacing between adjacent scan lines, equal to approximately 279.9nm. The gradient function uses second order central

'All surface data processing was done using MATLAB R2017a (Version 9.2.0.538062), MathWorks Inc., Natick, MA.

71 differencing at interior points, and first order forward or backward differenc- ing at boundary points, to compute gradients in the x and y directions. An important roughness parameter, that often appears in the calculation of free energy changes during wetting, is the Wenzel roughness rw; it is defined as the ratio of the developed area of the surface to its apparent (or projected) area. The Wenzel roughness is calculated from the local surface gradient using rw = V1 + Vh(x, y) 2 dA (3.14)

In MATLAB, the area integral on the discrete data set is approximated by the trapezoidal rule using nested trapz functions:

rw = trapz(Y, trapz(X, sqrt(1 + Gx.^2 + Gy.^2), 2)) / A;

X and Y are, respectively, vectors containing the x and y coordinates of the scan lines, and A is the total scan area. The value of rw computed this way depends on the spatial resolution of height data; for a truly self-affine fractal surface with roughness features at all wavelengths, the integral in equation (3.14) diverges [90]. The finite resolution of the profilometer, how- ever, limits the range of measurable wavelengths, and roughness features smaller than the grid spacing c are inevitably excluded from the measured height profile. This absence of high frequency roughness components limits the usefulness of the value of rw calculated from discrete data; it may still be used as a representative value for qualitative comparisons of wetting be- havior between surfaces, but is not sufficiently accurate to be used directly in surface free energy calculations. Besides the mean square roughness, we can also compute higher moments of the height distribution. The skewness -y and the kurtosis #3are defined as the standardized third and fourth moments of profile heights:

(h (x, y))A (h(x,y))A (315) w 3 w 4 The interface width, skewness and kurtosis provide information about the vertical distribution of roughness elements about the mean plane. To obtain a measure of the horizontal or lateral length scale of surface features, we first compute the two-dimensional autocorrelation function [90]:

R(u, v) = 2 (h(x + u, y + v) - h(x, Y)),, (3.16) W

72 The variables u and v are the spatial lags in the x and y directions. The auto- correlation function R(u, v) gives the degree of correlation between roughness elements that are separated by a distance u in the x direction and v in the y direction. For the discrete height data, we use MATLAB's xcorr2 function to get a biased estimate of R(u, v): R = xcorr2(h)./(w^2 * 1024 * 768); In plane polar coordinates (r, 0), the autocorrelation function is given by R(r, 0) = R(r cos 0, r sin 0) (3.17) From equations (3.12) and (3.16), R(r, 0) = 1 when r = 0; at large values of r, the autocorrelation function R(r, 0) typically decays to zero, since on a random texture the surface heights at large separations are uncorrelated. The autocorrelation length A0 is defined as the value of the radial lag r, along the 0 direction, at which the autocorrelation function first decays to 1/e:

A = inf {r I R(r, 9) < 1/e} (3.18)

The smallest and largest autocorrelation lengths are then, respectively,

A = inf{ Ao 0 < 27r} and A 2 = sup {Ao 10 < 0 < 2r}

The ratio of A, to A2 is a measure of the anisotropy of the rough surface, and is called the texture aspect ratio: A= (3.19) A2 For an isotropic surface, the autocorrelation function is radially symmetric, and therefore A = A2 giving = 1. Highly anisotropic surfaces have values of much smaller than 1, and a one-dimensional surface profile has a texture aspect ratio of 0. For surfaces that are not highly anisotropic, as is the case with most random rough surfaces, a mean autocorrelation length A can be calculated by averaging the value of A0 in different directions:

A = (A) 0 (3.20) The mean autocorrelation length A gives a characteristic length scale for the lateral separation of roughness elements; surface features separated by distances larger than A are, on average, uncorrelated [90]. In MATLAB, the mean autocorrelation length (stored in variable lambda) was computed from the discrete autocorrelation matrix R using the following code:

73 B = bwperim(R >= exp(-1)); u = (-1023 1023) * c; v = (-767 767) * c; [U, V] = meshgrid(u, v); lambda = mean(sqrt((U(B)).^2 + (V(B)).^2)); % Mean ACL

A simple model for a randomly generated rough surface is the Gaussian surface 191,92], in which the height h(x, y) is normally distributed about the mean plane (here, z = 0). The probability density function for a Gaussian surface with interface width w is given by 1 ( z2 P (z < h < z + dz) = exp dz (3.21) 27rw 2 / A measure of the departure of the height distribution from the ideal Gaussian function is the normalized excess kurtosis, defined as:

K - 3 (3.22) 3 The normal distribution has a kurtosis of 3, and therefore r = 0 for a Gaus- sian surface; non-zero values of r denote deviations from the ideal Gaussian distribution. For a random Gaussian surface, the gradients in the x and y directions are also normally distributed [91-93]:

1 q2 P(q< hx< q+dq)= exp q dq V2_ si 2S2 1 q 2 1 P(q

rw =1 + exp - erfc - (3.23) 2 s2 S

For large values of s, the expression on the right hand side is well approxi- mated by

rw ~ (3.24) 2

74 Therefore, a random superhydrophic texture with a large mean square slope generally also has a large Wenzel roughness, and is more resistant to pressure induced wetting [64]; this connection between surface slope and interface stability is discussed in more detail in @4.1.

3.2.3 Contact angle goniometry The apparent contact angle of a sessile water drop on each of the three su- perhydrophobic test surfaces was measured using a contact angle goniometer (Model 590 goniometer, ram6-hart instrument co., Succasunna, NJ). The ad- vancing and receding contact angles were measured using the drop volume method. A drop of deionized water, attached to a stainless steel dispensing needle (SAE 304, 22 gauge, outer diameter 0.711 mm), was deposited gen- tly on the test surface. The volume of the drop was gradually varied by first introducing and then withdrawing liquid through the needle, and the apparent contact angles on either side of the drop profile were measured as the basal contact line advanced or receded on the substrate. During mea- surements, the drop volume was kept between 10-20 pL, corresponding to an excess Laplace pressure of 87-109 Pa within the drop. To obtain quasi-static measurements, the drop volume was increased and decreased at a slow rate of 0.25 pL s-1. The contact angles were estimated by fitting a circular arc to the drop profile. The advancing and receding angles on each of the test surfaces were measured at two or three different spots and then averaged; the final values are listed in Table 3.3.

3.3 Rough superhydrophobic surfaces

Three different types of rough superhydrophobic surfaces were investigated in this study, as potential candidates for passive drag reduction in turbu- lent flows. The first of these was a hydrophobic polymer coating applied to the substrate, with the requisite surface roughness generated by an additive, spray-coating process. In the second case, the rough texture was produced on a metallic substrate by a subtractive, acid etching process. Both spray- coating and chemical etching provide scalable methods to generate a uniform, rough texture over large areas of the base substrate. The third surface had a hierarchical texture, with roughness features at multiple length scales pro- duced by three successive texturing processes - sandblasting, chemical etch-

75 Table 3.3: Experimentally measured surface statistics and apparent contact angles for rough superhydrophobic test surfaces. The reported values were obtained by averaging measurements at two or three different spots on each surface.

Parameter Acrylic FPOSS Etched Sandblasted, etched, spray coating aluminum and boehmitized aluminum 80 grit 150 grit

Root mean square roughness w (pm) 11.02 6.505 3.853 3.359 Root mean square slope s 4.539 2.955 2.348 2.946 Wenzel roughness rw 3.242 2.330 2.282 2.697 Skewness 'y 1.635 -0.044 -0.059 -0.219 Kurtosis 3 5.369 3.105 3.163 2.738 Norm. excess kurtosis K 0.790 0.035 0.054 -0.087 Mean autocorrelation length A (pm) 15.18 20.43 12.29 7.781 Texture aspect ratio 0.817 0.855 0.851 0.906 Advancing contact angle Oa 1590 + 10 1620 10 1520 10 1510 t 10 Receding contact angle or 1570 t 10 1580 t 10 1480 10 1480 10 ing and boehmitization. The fabrication and experimental characterization of these surfaces are described in the following sections.

3.3.1 Acrylic FPOSS spray coating Air assisted spray coating techniques can be used to deposit drag-reducing superhydrophobic textures quickly and efficiently over large areas of the base substrate. Srinivasan et al. (2011) developed a sprayable superhydrophobic coating consisting of poly(methyl methacrylate) and 1H,1H,2H,2H-heptadeca fluorodecyl polyhedral oligomeric silsesquioxane (fluorodecyl POSS), both dissolved in a volatile perfluorinated solvent [36]; the polymer helps the coat- ing adhere to the substrate, and the spray atomization process generates the requisite surface roughness. Chemical hydrophobicity is imparted by the fluoroalkylated POSS (or FPOSS) molecules, each having a silicon-oxygen polyhedral cage attached to eight highly fluorinated alkyl side chains [94]. Because of these fluoroalkyl groups, FPOSS crystals have very low values of solid surface energy; in fact, fluorodecyl POSS is one of the most hydropho- bic crystalline solids known, with a surface energy USA ~~10 mN m-1 and a water contact angle of about 1250 [95,96]. The surface morphology of the coating can be controlled by varying the molecular weight of PMMA used, or its concentration in the solution; a corpuscular structure is obtained with dilute solutions, whereas at higher concentrations, extensional viscosity of the polymer impedes fragmentation of spray filaments, producing a beads- on-string or fibrous structure [36]. The roughness of the polymer matrix and the presence of low surface energy FPOSS moelcules on the surface result in excellent water repellency of the coating and low contact angle hysteresis. The original formulation of the coating was slightly modified before use in the present study. PMMA of a lower molecular weight was used, and the more readily available 1-I,1H11,211,2H-tridecafluorooctyl polyhedral oligomeric silsesquioxane (fluorooctyl POSS) was substituted for fluorodecyl POSS. Flu- orooctyl POSS has a slightly higher surface energy than fluorodecyl POSS and is therefore less effective; it is, however, commercially available and was easier to obtain in large quantities, unlike fluorodecyl POSS which has to be synthesized in the lab by condensing the corresponding trialkoxysilane.

Surface preparation. Poly(methyl methacrylate) powder, of molecular weight 75 kDa, was purchased from Scientific Polymer Products Inc., Ontario, NY; fluorooctyl POSS was supplied by NBD Nanotechnologies Inc., Boston, MA.

77 Figure 3.1: A depth composed optical micrograph of the acrylic FPOSS spray coating, showing the corpuscular microstructure of the poly(methyl methacrylate) matrix. The scale bar corresponds to 50 pm.

To prepare the spray solution, 25 g L- 1 of PMMA and 25 g L-1 of fluorooctyl POSS were dissolved in Asahiklin AK-225, 2 a commercial hydrochlorofluoro- carbon solvent manufactured by Asahi Glass Company, Alpharetta, GA. Before spraying, the rotor surface was cleaned by sonicating in a bath of acetone, and subsequently rinsing with isopropanol and deionized water. The spray solution was then applied evenly on the surface with a hand-held airbrush (Iwata HP-BC plus, nozzle diameter 0.3 mm), using compressed nitrogen gas at 50 psi for atomization. The nozzle was held at a distance of 20 cm from the substrate throughout the spraying process. Only the outer cylindrical surface was coated, whereas the stem and top flat of the rotor were left untreated and hydrophilic. After spraying, the coating was allowed to dry in ambient air at room temperature for 3 h. The superhydrophobic surface obtained by this method had a corpuscu- lar microstructure, as seen in Figure 3.1. The surface had excellent water repellency (0a = 159', 0, = 1570) and produced a uniform reflective plastron when immersed in water. The coating was, however, mechanically fragile and required careful handling during tests, and was easily damaged by abrasion or contact with other surfaces.

2AK-225 consists of the isomeric halocarbons 3,3-dichloro-1,1,1,2,2-pentafluoropropane (HCFC-225ca) and 1,3-dichloro-1,1,2,2,3-pentafluoropropane (HCFC-225cb).

78 ' I I I I 1$ 30 / 7 / 7 7 7 28 7

$-4 7 0 7A 7 -- 9-- & 7 .- 26 7 4 0 7 9- 7 9-

24 ~ $.-4 ;47 7 - 7 A -@~~ 7 22 0 -7' 0

20 I i 1000 2000 3000 Shear Reynolds number Re*

Figure 3.2: Skin friction data for the acrylic FPOSS spray coating in Prandtl- von Kdrmin coordinates. The black circles ( 0 ) represent baseline measurements on a smooth aluminum rotor. The red triangles ( A ) denote experimental data for the acrylic FPOSS spray-coated rotor. The black line (--) is the baseline friction curve for the TCL device, given by equation (2.29). The red curve (--) is a least-squares fit of equation (3.10) over 7 data points, giving a slip length of b = 11.3 pm.

~" C 20

a, 10 A A A A A A A

a, Q 0 5.-I a, 1 2 3 4 5 6 Reynolds number Re .104

Figure 3.3: Percentage drag reduction as a function of the Reynolds number Re for the acrylic FPOSS spray-coated surface.

79 Friction measurements. Skin friction measurements on the spray-coated rotor were performed in the TCL device, with deionized water as the working fluid. Figure 3.2 shows the experimental data in Prandtl-von Kirmin coordinates. The surface gave a slip length of b = 11.3 pm, comparable to the value of 12 pm measured by Srinivasan et al. on the original PMMA and fluorodecyl POSS coating using the TCO device 160]. Figure 3.3 shows the percentage drag reduction as a function of the flow Reynolds number Re; a maximum drag reduction of 7.9 % was obtained at Re = 34203. Visual inspection of the rotor showed that the coating underwent consid- erable mechanical wear during the test, with some of the sprayed material scoured off the surface by the flow and deposited on the stator walls. Al- though the coating remained hydrophobic when reimmersed in water, the plastron formed was dull and uneven, unlike the smooth and reflective plas- tron observed on the pristine coating at the beginning of the test. A second, subsequent flow test was performed on the same rotor after it was allowed to dry completely in air; no drag reduction (or drag increase) was observed, suggesting that damage to the surface was irreversible.

Surface characterization. Surface characterization tests were performed on a flat glass slide spray-coated under identical conditions as the rotor. Fig- ure 3.4 shows a representative height profile of the spray-coated surface. obtained using the laser profilometer. Profile statistics and contact angles were measured at two different spots on the surface and averaged; these are listed in Table 3.3.

3.3.2 Etched aluminum Surface preparation. The etched aluminum surface was prepared by collabo- rators at the University of Michigan, using a subtractive acid-etching process based on the surface modification technique previously reported by Yang et al. (2011) 138]. The rotor, made of aluminum 6061-T6 alloy, was first cleaned by ultrasonication in acetone, followed by rinsing with isopropanol and wa- ter. It was then etched in 2.5 M hydrochloric acid for 20 min to generate the rough surface microtexture. An inadvertent consequence of acid etching was the appearance of a faint longitudinal etch pattern on the rotor surface, caused by preferential corrosion along grain boundaries of the original ex- truded stock from which the rotor was machined. After etching, the residual metal particles were removed by sonication, and the surface was immersed

80 r4

50

'S 0 200 -5% 150 50 10100 150 (0n 250 0 y (Im) X (Jim)

-10 0 10 20 30 40 50

Figure 3.4: 2D profilogram of the acrylic FPOSS spray coated surface. The height data has been downsampled by a factor of 16 for rendering. in boiling deionized water for 20 min. The native oxide layer dissolves in boiling water and redeposits as boehmite, a mineral form of aluminum oxy- hydroxide with rhombohedral bipyramidal crystals [39]. The boehmite grows as a conformal layer of crossed nanoscale leaflets, each 50-100 nm long and 10-30 nm in thickness, providing an additional level of roughness over that produced by etching. Finally, the surface was hydrophobized by vapor phase deposition of 1H,1H,2H,2H-heptadecafluorodecyl triethoxysilane at 80 C un- der vacuum. The resulting superhydrophobic texture had good mechanical strength and excellent water repellency (Oa = 1620, Or = 1580), entraining a uniform reflective plastron when immersed in water. Figure 3.5 shows a depth-composed optical image of the final surface, and the microscale fea- tures produced by the etching process are clearly visible.

Friction measurements. Flow tests were performed in the TCL device, first with deionized water, and then with synthetic seawater; Figure 3.6 shows friction data from both tests in Prandtl-von Karman coordinates. A slip

81 Figure 3.5: A depth composed optical micrograph of the etched aluminum surface. The scale bar denotes 50 pm. length of b = 32.1 pm was obtained in the first test with water; the second test with seawater yielded a smaller value of b = 22.8 pm, most likely as a result of minor surface wear sustained during the previous test. The percentage drag reduction is plotted against the flow Reynolds number Re in Figure 3.7. The maximum drag reduction measured in the two tests were, respectively, 26.1 % in water, at Re = 59 408, and 16.1 % in seawater, at Re = 56 604. In both tests, the plastron was gradually sheared off by the flow at higher speeds, causing the decrease in percentage drag reduction at large Reynolds numbers seen in Figure 3.7.

Surface characterization. Surface characterization tests were performed on a flat aluminum substrate processed under identical conditions as the rotor surface. Figure 3.8 shows a representative height profile of the etched surface, obtained using the laser profiloieter. Profile statistics and contact angles were measured at two different spots on the flat substrate and then averaged; the final values are reported in Table 3.3.

3.3.3 Sandblasted, etched and boehmitized aluminum The stability of the air-water interface on a superhydrophobic surface, and the resistance of the texture to pressure-induced wetting, can be increased by incorporating hierarchical roughness features at two or more length scales [971. A three tier hierarchical superhydrophobic surface was prepared for the

82 32 1 1 1 A (j 30

228

4-D26 -A 4 7

S240 - ~ i AC 4-D22Ol 0

20 IIi i * I 1000 2000 3000 Shear Reynolds number Re*

Figure 3.6: Skin friction data for the etched aluminum surface in Prandtl-von Kdr- min coordinates. The filled ( e ) and empty ( o ) black circles denote, respectively, baseline measurements on a smooth rotor in water and synthetic seawater; the red ( A) and blue ( + ) symbols are the corresponding data points for the etched rotor. The black line (--) is the baseline friction curve for the TCL device, given by equation (2.29). The red (--) and blue (--) curves represent least-square fits of equation (3.10) to the data, giving b = 32.1 pm in water (first 8 data points) and b = 22.8 pm in synthetic seawater (first 7 data points). present study by the collaborating team at the University of Texas at Dal- las. Three levels of roughness features at successively smaller length scales were produced on the aluminum surface by sandblasting (large features of the order of 100 pm), acid etching (secondary features of size 1-10 pm), and boehmitizing (tertiary features of size 10-100 nm); the textured substrate was then hydrophobized to obtain a robust and mechanically durable super- hydrophobic surface.

Surface preparation. Two identical aluminum rotors were sandblasted with abrasive grits of different mesh sizes - 80 grit alumina (with 165 pm average particle size) was used on one rotor, and 150 grit alumina (with 89 pm av-

83 30 A A A A 0 20- AA A 10 -

Q 0 II I I I I I 1 2 3 4 5 6 7 8 9 Reynolds number Re .1

Figure 3.7: Percentage drag reduction as a function of the Reynolds number Re for the etched aluminum surface, in water ( A) and synthetic seawater ( + ).

'1r 1

25 0

150 200 50 y (pm) x (Jim) 250 0

I -AMMI -15 -10 -5 0 5 10 15

Figure 3.8: 2D profilogram of the etched aluminum surface. The height data has been downsampled by a factor of 16 for rendering.

84 (b) (C)

Figure 3.9: Scanning electron micrographs of the 80 grit sandblasted, etched and boehmitized surface. (a) Hierarchical surface texture before hydrophobiza- tion, with roughness features at multiple length scales. (b) The boehmite layer before hydrophobization, showing the characteristic crossed leaflet nanostructure. (c) After treatment with Glaco solution, the boehmite is completely covered by fluorinated silica nanoparticles. The scale bar in (a) denotes 10 pm; those in (b) and (c) denote 1 pm. Images were taken using a Zeiss Ultra Plus field emission SEM at the Center for Nanoscale Systems, Harvard University.

85 erage particle size) was used on the other. Both rotors were then etched in hydrochloric acid for 25s, and boehmitized in boiling water for 30 min. After texturing, the rotors were returned to MIT for hydrophobization. The textured rotor surfaces were made hydrophobic by treating with Glaco Mirror Coat Zero, a transparent anti-rain coating manufactured by SOFT99 Corporation, Japan; it consists of 30-50 nm fluorinated silica par- ticles dispersed in isopropanol [98], and is intended for use on automobile rearview mirrors. Before applying the Glaco solution, the rotors were cleaned by sonication in acetone for 2 min, followed by rinsing with isopropyl alcohol. Afterwards, they were placed in a plasma cleaner (PDC-001, Harrick Plasma, Ithaca, NY) and exposed to air plasma (at 30 W RF power and 200 mTorr chamber pressure) for 3 min to remove all residual contaminants. A uniform layer of Glaco solution was then sprayed on the surface, and allowed to dry completely in ambient air for approximately 5 min; after drying, the surface was baked on a hot plate at 250 C for 30 min. The solution spraying and baking steps were repeated a total of three times to get a robust and uniform superhydrophobic coating on the surface. Figure 3.9 shows scanning electron micrographs of the 80 grit surface both before and after the hydrophobiza- tion process; the porous boehmite layer helps retain the fluorinated silica particles in the texture, and is seen to be almost completely covered by a layer of particles after three rounds of coating. After hydrophobization, both rotors produced a uniform, reflective plastron when immersed in water.

Friction measurements. Flow tests were performed on both rotors in the TCL device with deionized water as the working fluid. Figure 3.10 shows the experimental data from the two tests in Prandtl-von Kdrmin coordinates. The 80 grit surface displayed a higher slip length of b = 23.6 Jim, compared to the 150 grit surface which gave a slip length of b = 8.4 pm. Figure 3.11 shows the percentage drag reduction as a function of the Reynolds number Re for the two rotors. The maximum drag reduction obtained in the two tests were, respectively, 21.3% (at Re = 59408) for the 80 grit surface, and 7.1 % (at Re = 41119) for the 150 grit surface.

Surface characterization. For surface characterization tests, two flat aluminum substrates were processed under identical conditions as the rotor surfaces. Figure 3.12 shows representative height profiles for the 80 and 150 grit sur- faces, obtained using the laser profilometer. For each surface, profile statistics

86 32 I I /

q30 - -

A q 28 -

0 4- 26 loo7

S24-7 '

42P 22 0

20 1000 2000 3000 Shear Reynolds number Re*

Figure 3.10: Skin friction data for the sandblasted, etched and boehmitized alu- minum surfaces in Prandtl-von Kirmdn coordinates. The black circles ( 0 ) denote baseline measurements on a smooth rotor. The red (,A ) and blue (+ ) symbols are, respectively, experimental data for the 80 grit and 150 grit surfaces. The black line (--) is the baseline friction curve for the TCL device, given by equation (2.29). The red (- -) and blue (--) curves are least-square fits of equation (3.10) to the data, giving b = 23.6 jim for the 80 grit surface (first 9 data points) and b = 8.4 pm for the 150 grit surface (6 data points).

87 30

20 - A A A A AA c~10 - A A

1 2 3 4 5 6 7 8 9

Reynolds number Re .104

Figure 3.11: Percentage drag reduction as a function of the Reynolds number Re for the 80 grit ( A ) and 150 grit ( + ) sandblasted, etched and boehmitized aluminum surfaces. and contact angles were measured at three different spots and then averaged; the final values are listed in Table 3.3.

3.4 Streamwise grooved surfaces

Drag reduction in turbulent flows can also be achieved using an array of streamwise grooves infused with a secondary fluid, which in effect produces a pattern of alternating slip and no-slip stripes on the wall [85,86]. The lubricating fluid inside the grooves can be either air, as in the case of two- dimensional superhydrophobic grooves, or a liquid that preferentially wets the. grooves and is immiscible with water, as in the case of liquid-infused grooves. Computational studies of wall-bounded turbulent flows show that streamwise slip reduces drag, whereas spanwise slip tends to intensify near- wall coherent structures, leading to enhanced turbulence production and an increase in frictional drag [52]. Insofar as streamwise grooves have a smaller slip length in a direction transverse to the flow, they have a potential ad- vantage over randomly rough superhydrophobic surfaces, which are usually isotropic and have identical slip lengths in both the streamwise and spanwise directions. In addition to the three superhydrophobic surfaces described in the previous section, two types of streamwise grooved surfaces were also char- acterized as part of this study - one with the grooves filled with air in the Cassie-Baxter state, and the other with the grooves completely filled with a low viscosity alkane. Air has a low viscosity and provides an almost shear-

88 (a) 80 grit

-1=

25 0 I 200 -2 ) 150 50 1001 00 150 200 50 y (Irm) x (Pmn) 250 0 (b) 150 grit

25 0 1200

150 200 50 y (Pm) x (PM) 250 0

I I -15 -10 -5 0 5 10 15

Figure 3.12: 2D profilograms of the (a) 80 grit and (b) 150 grit sandblasted, etched and boehmitized aluminum surfaces. The height data has been downsampled by a factor of 16 for rendering.

89 Figure 3.13: The streamwise grooved rotor after deposition of the organosilane monolayer. Droplets of water (dyed blue) do not wet the rotor surface after the silanization process. free interface, but is also more readily displaced from the grooves; the alkane lubricant, on the other hand, generates less slip owing to its higher viscosity, but has a greater resistance to shear-induced drainage under turbulent flow conditions [85].

3.4.1 Liquid-infused streamwise grooves

Rosenberg et al. (2016) obtained turbulent drag reduction inside a Taylor- Couette apparatus3 by machining external threads on the inner cylinder and filling the thread grooves with a low viscosity alkane or perfluorinated oil 185]. Similar tests were performed in the TCL device, on a rotor with regularly spaced circumferential grooves filled with heptane.

Surface preparation. An aluminum rotor with streamwise grooves was ma- chined by the collaborating team at Princeton University, for use in flow tests in the TCL device. Square grooves 200 pm deep and 200 pm wide were ma- chined at regular 400 pm intervals on the rotor surface using a precision CNC lathe; Figure 3.17(a) shows an SEM image of the finished groove profile. The rotor was then shipped to MIT for surface treatment and flow tests.

3 The TC apparatus used in their study had a rotating outer cylinder and a stationary inner cylinder; the frictional torque was measured on the inner cylinder.

90 Table 3.4: Physical properties of n-heptane at 25 'C. The surface tensions with and without the azodye were measured experimentally by the du Noiiy ring method; all other values were obtained from standard reference tables [99].

Property Value Density kg m- 3 6.796 x 102 0.682 p(water) Dynamic viscosity Pas 3.88 x 10-4 = 0.436 -p(water) Kinematic viscosity m 2 s-1 6.07 x 10- = 0.680 v(water) Vapor pressure Pa 6.091 x 103 Vapor pressure (-12 C) Pa 6.973 x 102 Surface tension (pure) N m 1 2.0045 x 10-2 Surface tension (with dye) N m- 1.9875 x 10-2

At MIT, the rotor was thoroughly cleaned with a detergent solution to remove residual swarf and machining oil inside the grooves. It was then sonicated in acetone for 30 min, and rinsed afterwards with isopropanol and water. Before further surface treatment, a control test was performed in the TCL device with no air or lubricant inside the grooves; the rotor was first exposed to oxygen plasma (30 W RF power, 200 mTorr chamber pressure) for 15 min to make the surface strongly hydrophilic, so that water would wet and completely fill the grooves during the test. To prepare the grooves for alkane infusion, the rotor was boehmitized in boiling water for 30 min, and allowed to dry completely in ambient air for several hours. After all the water had evaporated from the grooves, the rotor was immersed in a 1 % solution of trichloro(octadecyl)silane (Sigma- Aldrich) in anhydrous n-heptane for 8 h. The grooves were subsequently rinsed with pure n-heptane to remove any unreacted silane, and then dried in a compressed nitrogen stream. The organosilane monolayer increases the affinity of the surface for the infused alkane, and helps retain the lubricant in the grooves during flow tests; it also renders the surface hydrophobic, as seen in Figure 3.13.

Friction measurements. Flow tests were performed in the TCL device with water as the working fluid, and n-heptane as the lubricant inside the rotor grooves. Pure heptane at room temperature is a colorless and volatile liquid, with a lower density and viscosity than water; the properties of n-heptane at 25 'C are listed in Table 3.4. Because of its high volatility, the heptane

91 30

,/A, 28 A"/

0 V7 V V V7 26 7 0 Q '17 I- 24

22 0 0

20 1000 2000 3000 Shear Reynolds number Re*

Figure 3.14: Skin friction data for liquid-infused streamwise grooves in Prandtl- von Kdrmin coordinates. The black circles ( 0 ) represent baseline measurements on a smooth rotor. The red triangles ( A ) denote data for heptane-filled grooves, and the green inverted triangles ( v ) denote data from the control experiment with hydrophilic, water-filled grooves. The black line (--) is the baseline friction curve for the TCL device, given by equation (2.29). The red curve (- -) is a least-squares fit of equation (3.10) over the first 7 data points, giving b = 11.0 pim.

20

I- 10 A A A A A

0

1 2 3 4 5 6 7 8 9 Reynolds number Re _104

Figure 3.15: Percentage drag reduction as a function of the Reynolds number Re for streamwise grooves infused with n-heptane.

92 (a) Before test (b) After test

Figure 3.16: The heptane-infused grooved rotor, imaged before and after flow tests. Several grooves have failed by the end of the test, and the displaced heptane is visible as a thin red layer of liquid floating at the top of the cell. was cooled down to (-12 3) 'C before infusion into the grooves; the vapor pressure falls by almost 89 % at this temperature, and evaporation from the grooves during the filling process is significantly reduced. To ensure that all grooves were completely filled, a surplus amount of cold heptane was pipetted onto the surface of the rotor, and the excess was allowed to drain off under gravity. The rotor was then quickly lowered into the pre-filled stator, containing water at 25*C; all grooves were thus fully under water within 1 min of filling, preventing further evaporation of the lubricant. The actual flow test was started after an additional 5 min, to allow time for the heptane in the grooves to warm up to the test temperature. To visually monitor the drainage of lubricant from the grooves, approxi- mately 0.1 g/L of Oil Red EGN, a water-insoluble red azodye, was dissolved in the heptane to make it visible during tests. To ensure that the addi- tion of dye does not adversely affect the interfacial tension of heptane with water, the surface tension of both pure heptane and the dye solution were measured separately with a dynamic tensiometer (DCAT 11, DataPhysics GmbH, Germany) using the du Noiiy ring method. The surface tension val- ues are reported in Table 3.4; no significant change in surface tension was caused by the presence of the dye. Figure 3.14 shows the experimental data from flow tests in Prandtl- von Karman coordinates. The control test with water-filled grooves showed

93 an increase in drag, which is consistent with previous computational stud- ies in the literature [100]. A slip length of b = 11.0 pm was obtained for the heptane-infused grooves, which is comparable to that obtained for the acrylic FPOSS spray coating. Figure 3.15 shows the percentage drag reduction as a function of the flow Reynolds number Re for the lubricant-infused surface; a maximum drag reduction of 7.9% was measured at Re = 34203. With further increase in rotor speed, the lubricant was gradually stripped off the grooves by the flow, as seen in Figure 3.16; a significant fraction of grooves had failed by the end of the test, with the displaced heptane floating to the top of the flow cell. The rotating inner cylinder configuration of the TCL device aids in the retention of lubricant, since heptane is less dense than water and is forced inwards to the rotor surface by centrifugal effects. This artifact of the flow geometry should be taken into account when interpreting the results of flow tests; in real-life service, the lubricant may be displaced from the grooves at a smaller shear stress than observed in flow experiments.

3.4.2 Air-filled streamwise grooves Surface preparation. When flow tests on the heptane-infused surface were completed, the grooved rotor was thoroughly cleaned by ultrasonication in heptane to remove all dye residue from the surface, followed by rinsing with acetone, isopropanol and water. The organosilane layer on the surface was removed by exposing the rotor to oxygen plasma (30 W RF power, 200 mTorr chamber pressure) for 10 min. The clean, hydrophilic rotor was then sent to the collaborating team at Pennsylvania State University for further surface modification; there, the rotor was boehmitized and refunctionalized with (heptadecafluoro-1,1,2,2-tetrahydrodecyl)trichlorosilane, to generate chemi- cal affinity for the perfluorinated lubricant which would be infused later into the nanotexture to produce a 'slippery rough' surface. Figure 3.17 shows scanning electron micrographs of the grooved surface, imaged after the sur- face treatment. The rotor was shipped back to MIT for flow testing. Immediately before flow tests, a synthetic perfluoropolyether oil (DuPont Krytox GPL 102) was pipetted onto the rotor surface until all grooves were completely wetted by the lubricant. The oil inside the groove troughs was then completely flushed out using a high pressure nitrogen gun, leaving only a thin layer of lubricant imbibed in the boehmite nanotexture, conformally coating the groove profile. The coating of perfluorinated oil imparts hy-

94 Figure 3.17: Scanning electron micrographs of the air-filled streamwise grooved surface, before application of the perfluoropolyether lubricant. (a) Grooves on the rotor. The scale bar denotes 200 pm. (b) Magnified view, showing the boehmite nanotexture; patches of polymerized organosilane are visible. The scale bar denotes 1 pm. Images were provided by Jing Wang, Department of Mechanical and Nuclear Engineering, Pennsylvania State University, and were obtained using a Zeiss Merlin field emission SEM at the Nanofabrication Laboratory of the Materials Research Institute at PSU. drophobicity to the surface, so that air is retained in the grooves when the rotor is submerged under water, as seen in Figure 3.20(a). Furthermore, at the groove crests, the no-slip contact between water and the solid substrate is replaced by a 'slippery' partial-slip contact between the lubricant layer and water.

Friction measurements. Flow tests were performed in the TCL device, with both deionized water and synthetic seawater as working fluids. Figure 3.18 shows friction data from both tests in Prandtl-von Kairmin coordinates. A slip length of b = 17.9 pm was obtained with water, and a comparable value of b = 15.4 pm was obtained with seawater. The percentage drag reduc- tion as a function of the flow Reynolds number Re in shown in Figure 3.19. The maximum drag reduction measured in the two fluids were, respectively, 12.5% in water, at Re = 34 203, and 11.6% in seawater, at Re = 47102. In both tests, the air inside the grooves was sheared away by the flow at higher

95 30

~ A A 28 - -

26 - - A

22 - V

20 1000 2000 3000 Shear Reynolds number Re*

Figure 3.18: Skin friction data for air-filled streamwise grooves in Prandtl- von Kdrmdn coordinates. The filled ( 0 ) and empty ( o ) circles are, respectively, baseline measurements on a smooth rotor in water and synthetic seawater; the red ( A) and blue ( + ) symbols are the corresponding data points for air-filled grooves. The green symbols ( v ) denote data from the control test with water-filled grooves, showing drag increase. The black line (- -) is the baseline friction curve for the TCL device, given by equation (2.29). The red (- -) and blue (- -) curves repre- sent least-square fits of equation (3.10) over the first 7 data points in each series, giving b = 17.9 pm in water and b = 15.4 prn in synthetic seawater.

96 2 0 I I I I I I

10y mbA +AA

1 2 3 4 5 6 7 8 9 Reynolds number Re _i10

Figure 3.19: Percentage drag reduction as a function of the Reynolds number Re for air-filled streamwise grooves, in water ( A ) and synthetic seawater (+). speeds, with a concomitant drop in the percentage drag reduction at large Reynolds numbers. At the end of each test, several grooves were observed to have transitioned to a water-filled Wenzel state, as seen in Figure 3.20. Because of its low surface tension and high viscosity, the perfluorinated lubricant was not removed from the surface by the flow, and therefore did not have to be replenished between test runs; in fact, no appreciable deterioration in surface quality was observed even after several successive tests with the same rotor. Flow tests were also performed with two other lubricants of lower viscosity - Krytox GPL 101 and 100 - and very similar slip lengths were obtained in all cases; as expected, the slip length of air-filled grooves was not dependent on the viscosity of the oil used. Furthermore, the complete purging of lubricant inside the grooves was found to be critical for achieving drag reduction; markedly low levels of drag reduction, and sometimes drag increase, were observed when the grooves were either partly or completely filled with oil.

3.5 Summary of experimental results

Table 3.5 summarizes the results of flow tests on all drag-reducing surfaces characterized in this study. Of the six surfaces tested in the TCL device, the largest slip length and percentage drag reduction were obtained in the case of the etched aluminum surface. The 80 grit sandblasted, etched and boehmi- tized aluminum rotor was the most resistant to failure under turbulent flow, and withstood a maximum shear stress of 26.9 Pa before the plastron was

97 (a) Before test (b) After test Figure 3.20: The air-filled streamwise grooved surface, imaged before and after flow tests. Air has been displaced from several grooves by the end of the test, especially near the top and bottom ends of the rotor.

sheared away by the flow. The 80 grit surface gave a much higher slip length than the 150 grit surface, suggesting that the length scale of the primary roughness features has a significant influence on slip length and percentage drag reduction in the case of hierarchical superhydrophobic textures. The surface easiest to fabricate was the acrylic FPOSS spray coating; however, it was also mechanically fragile, and was damaged irreversibly after a single test. The adhesion of the polymer base to the smooth metal surface of the rotor was found to be especially poor. All six surfaces failed during flow tests, well before the maximum wall shear stress of 50 Pa attainable in the TCL device. Figure 3.21 compares the drag reduction performance of all six model surfaces in terms of the maximum percentage drag reduction obtained, the effective slip length of the surface in turbulent flow, and the wall shear stress at which surface failure was observed. Among the two streamwise grooved surfaces, air-filled grooves gave a larger slip length than heptane-infused grooves, as is to be expected because of the lower viscosity of air compared to heptane. Interestingly, both air- filled and liquid-filled grooves failed during tests at comparable values of wall shear stress. Philip (1972) derived exact analytical expressions for the effective slip length b for different cases of viscous shear flow over a periodic array of alternating no-slip and no-shear stripes on the wall [46,471; in the

98 SEB 150 grit - -

LI grooves -

Acrylic FPOSS - AF grooves 12.5 SEB 80 grit - 21.3

Etched Al - 26.1 I I I ) 10 20 30 4 0 (a) Maximum DR (%)

SEB 150 grit - I////////28.4 LI grooves - ///////////211 Acrylic FPOSS - T//////////A11.3 AF grooves - MM 17.9 SEB 80 grit - 23.6

Etched Al - 32.1 I I I ) 10 20 30 4 0 (b) Slip length (pm)

SEB 150 grit - V//////////212.1- AF grooves - //// / 15.6 LI grooves - 16.5 Acrylic FP OSS - 7///////////2/216.5 Etched Al 18. 1 SEB 80 grit 26.9 0 10 20 30 40 (c) Failure shear stress (Pa)

Figure 3.21: Drag-reducing test surfaces ranked in the order of increasing (a) max- imum percentage drag reduction, (b) slip length and (c) wall shear stress at failure. All values are for tests with deionized water. Abbreviations - SEB: 'sandblasted, etched and boehmitized' aluminum, LI: liquid-infused' grooves and AF: 'air-filled' grooves.

99 Table 3.5: Summary of flow measurements on all drag-reducing test surfaces. All reported values are for tests in deionized water. The shorthand 'SEB' stands for 'sandblasted, etched and boehmitized'.

Surface Slip length Max. DR Failure Re* Failure T (Pm) (%) (Pa) Acrylic FPOSS spray-coat 11.3 7.9 1832.4 16.540 Etched aluminum 32.1 26.1 1915.1 18.066 SEB aluminum (80 grit) 23.6 21.3 2336.1 26.882 SEB aluminum (150 grit) 8.4 7.1 1565.6 12.073 Heptane-infused grooves 11.0 7.9 1832.4 16.540 Air-filled grooves 17.9 12.5 1778.8 15.586

case of stripes aligned in the flow direction, the slip length is given by

/ ira b -log sec -- (3.25) 7r 2

Here I is the spatial periodicity of the stripe pattern, and a is the fraction of the boundary over which the no-shear condition applies. For the grooved rotor used in this study, 1 = 400 pm and a = 0.5; assuming that the air-water interface over the groove troughs approximates a flat, shear-free boundary, equation (3.25) predicts a slip length of b = 44.1 lim for the air-filled stream- wise grooved surface. The experimentally obtained slip length of b = 17.9 Pm, although of the same order of magnitude, is only about 41 % of the ideal, theoretical value; the discrepancy is probably due to a small number of de- fective grooves that fail even at low rotor speeds, and more importantly, the presence of a smaller but finite spanwise slip that strengthens near-wall co- herent structures and increases the turbulent shear stress on the wall 152]. In the case of streamwise grooves filled with a secondary viscous liquid, an ex- pression analogous to equation (3.25) for the effective slip length b in Stokes flow was derived recently by Schdnecker et al. [501:

__ 1 sin (wra/2) ~i 4 / r p b = l log sec [-r + log + sin (/2) (3.26) 2 2aDp 1 - sin ( Ja/2)

Here a is the fraction of the boundary with liquid-lubricant contact, and P and -pare, respectively, the viscosities of the bulk fluid and the lubricant

100 inside the grooves. The non-dimensional factor D is a function of both the contact fraction a and the aspect ratio of the groove profile; for grooves that are at least as deep as they are wide, D ~ 0.3402 when a = 0.5. With heptane as the lubricant, equation (3.26) yields a theoretical slip length of b = 25.6 pm. The experimentally measured value of b = 11.0 pm is once again only 43 % of the theoretical value. The results of surface characterization and contact angle measurements on the four superhydrophobic test surfaces are listed in Table 3.3. The spray- coated surface had a much larger root mean square roughness and root mean square slope than the aluminum surfaces textured by sandblasting and acid etching techniques. The etched aluminum surface, which showed the largest slip length during flow tests, also had the largest mean autocorrelation length, indicating that the lateral spacing of roughness elements is a key determinant of slip length on rough superhydrophobic textures. The relationship between specific statistical measures of surface roughness, and the drag reduction performance of the superhydrophobic texture, is explored in detail in the next chapter.

101 102 Chapter 4

Conclusion

4.1 Designing optimal textures for drag reduction

In chapter 3, the fabrication and subsequent flow characterization of four su- perhydrophobic test surfaces - the acrylic FPOSS spray coating, the etched aluminum surface, and the two sandblasted, etched and boehmitized alu- minum surfaces - were discussed in detail; the effective slip length b obtained for each surface is reported in Table 3.5. The roughness characteristics of these surfaces were also measured using optical profilometry, and a number of surface statistical parameters were computed from the profile height data; these are summarized in Table 3.3. With this information in hand, and draw- ing from the results of previous theoretical, experimental and computational investigations in the literature, we can now attempt to relate specific rough- ness parameters to the drag reduction performance of randomly textured superhydrophobic surfaces in turbulent flow. In Figure 4.1, slip lengths for the four surfaces are plotted against dif- ferent statistical measures of surface roughness; specifically, we focus on the influence of four roughness parameters, namely, the root mean square rough- ness (or the interface width) w, the root mean square slope s, the Wenzel roughness rw, and the mean autocorrelation length A. The three aluminum surfaces have similar RMS roughness (3.4 pm < w < 6.5 1m), but display a large spread in slip length. The acrylic FPOSS spray coated surface, on the other hand, had a very large roughness (w = 11.0 pm), nearly twice that of the aluminum surfaces, and consequently yielded only a modest slip length, comparable to (and slightly larger than) the 150 grit sandblasted, etched and

103 40 Etched Etched 30 -

4D 80-SEB 80-SEB S- 20 - A-FP A-FP 150-SEB * 150-SEB * - 10 - S

0 I I 0 5 10 15 0 1 2 3 4 5 RMS roughness w (Jm) RMS slope s

40 I - I I Etched Etched 30 80-SEB - 80-SEB -- 20 A-FP 10 150-SEB 0 -150-SEBg / P --

'S U 0 1 2 3 4 5 0 5 10 15 20 25 Wenzel roughness rw Mean autocorr. length A (pm)

Figure 4.1: The experimentally determined slip length b for the four rough super- hydrophobic test surfaces, plotted against various roughness parameters computed from surface profilometry; the broken lines in the bottom right figure are only for visual reference. Abbreviations - A-FP: acrylic FPOSS spray coating, Etched: etched aluminum, 80-SEB: 80 grit sandblasted, etched and boehmitized aluminum, and 150-SEB: 150 grit sandblasted, etched and boehmitized aluminum.

104 SEB 150 grit /f 0.41 SEB 80 grit Etched Al 0.98 Acrylic FPOSS 1. 0 0.5 1 1.5 2 Scaled roughness w+

Figure 4.2: The root mean square roughness of the four superhydrophobic test surfaces, scaled by the viscous length 6, at failure. The shorthand SEB stands for 'sandblasted, etched and boehmitized' aluminum. boehmitized surface. This decrease in slip length at large values of surface roughness is consistent with previous experimental and theoretical studies in the literature [40,101-103]. In contrast to the ideal flat slip boundary often used in computational investigations, the actual liquid-air interface over a rough superhydrophobic texture spans a tortuous three-dimensional contact line formed by irregularly shaped roughness asperities, forming a complex undulating surface of constant mean curvature 1641; the Young-Laplace equa- tion is satisfied locally at every point on the interface, and the Young contact angle is maintained at the solid boundaries. Previous studies on the effect of interface deformation on drag reduction show that the slip length is largest when the interface is perfectly flat, and decreases with increasing deformation of the interface, both into and out of the flow region 1102,103]. Richardson (1973), discussing the applicability of the no-slip condition, showed that even on a surface that is locally shear free, the macroscopic boundary condition for viscous flow becomes effectively no-slip if the surface is made sufficiently rough [101]. Inasmuch as the root mean square roughness w dictates the scale of vertical fluctuations of surface heights, the value of w must be suf- ficiently small to ensure that the air-water interface, which forms the actual flow boundary, remains largely flat. Bidkar et al. (2014) measured frictional drag on several rough superhy- drophobic surfaces with different values of RMS roughness, produced using a proprietary thermal spray process, in turbulent flow inside a water tunnel for 1 x 106 < Re < 9 x 106. They observed that an appreciable reduction in drag was obtained only on surfaces having RMS roughness atleast an order of magnitude smaller than the thickness of the viscous sublayer, which extends

105 to a distance of about 5J, from the wall. They therefore proposed

w - = w+ < 0.5 (4.1) 6V as a necessary condition for successful drag reduction in turbulent boundary layer flow [40]. At larger values of roughness, the tall surface asperities projecting through the liquid-air interface interact with the viscous sublayer, obstructing the flow path and incurring form drag, which offsets the drag reduction generated by interfacial slip. The roughness criterion becomes particularly restrictive at high Reynolds numbers, when the thickness of the viscous sublayer becomes extremely small, of the order of a few micrometers or lower; the superhydrophobic texture may then simply act as hydrodynamic roughness, producing zero drag reduction or even drag increase. Figure 4.2 shows the wall-scaled root mean square roughness w+, computed at the point of surface failure, for the four superhydrophobic test surfaces. Whereas the three aluminum surfaces have w+ < 1.0, the acrylic FPOSS spray-coating has a scaled roughness of w+ ~ 1.6, which explains the relatively small value of slip length in spite of its excellent water repellency in air, and also the rapid deterioration of the coating observed after plastron failure. Excluding the data point for the acrylic FPOSS surface, whose poor per- formance is attributable to its large surface roughness, we notice a strong correlation between the slip length b and the mean autocorrelation length A. For fractal self-affine surfaces, the autocorrelation length represents the only lateral length scale of the texture, and is representative of the horizontal sep- aration between surface asperities [90]. To see this, we consider a ID random self-affine profile h(x), having a lateral correlation length A; a simple model for the autocorrelation function for this surface is the exponential model, given by

R(u) = exp - H] (4.2) where H is the called the roughness exponent, or-Hurst exponent, of the frac- tal profile [90]. For ease of mathematical analysis, we restrict our attention to surfaces that are self-similar, i.e, surfaces for which H = 1. The power spectral density (PSD) function C(k) for the roughness profile can then be

106 obtained in closed form as the Fourier transform of R(u):

_W2 +O k C(k) =7 J R(u) eiuk du

2 W j cR(u) cos (ku) du (4.3)

= exp [

Here k is the wavenumber of the roughness components with wavelength I = 2ir/k, and the second step follows from the fact that R(u) is even in u. The moments of the PSD function are given by: +00 MO =jC(k) dk = w 2 , (4.4)

m2 =j C(k) k2 dk = 2W2 and (4.5)

M4 = C(k) k 4dk = 12W2 (4.6)

It can be shown that for a random Gaussian surface, the average distance LP between adjacent peaks (summits), and the average distance L, between suc- cessive zero crossings of the profile, are both related to the ratio of moments of the PSD function [92,93]:

L =27r = -rA L = 9 -T 7rA (4.7) Vm4 3 m 2 f2 The autocorrelation length A thus gives the length scale of separation of sur- face asperities; in this respect, it is analogous to the spatial periodicity L for regularly patterned surfaces consisting of ridges or posts.1 Although the

1This is strictly true only in the case of self-affine surfaces, with a uniform fractal scaling behavior at all length scales. 'Mounded' surfaces, on the other hand, have an underlying characteristic wavelength A' introduced during the texturing process, with a corresponding peak in the PSD spectrum. For such surfaces, the wavelength A' is more representative the lateral spacing of asperities or 'mounds', whereas the autocorrelation length A gives the lateral size of these mounds [90]. The roughness spectra of each of the four surfaces in this case showed no single dominant wavelength, and more closely resembled a self-affine behavior.

107 available data points are too few to make a definitive conclusion, we note that a strong dependence of slip length on the separation between asperities agrees at least qualitatively with the theoretical results for regularly pat- terned surfaces. For instance, over a regular 2D array of posts with spatial periodicity L, the slip length scales as 151]:

b ~ (4.8)

The above scaling is derived in the limit of viscous creeping flow; in the limit of high Reynolds numbers, we instead have 155]

L1/362/ 3 -b ~~ (4.9) where 6, is the viscous length scale in turbulence. In either case, we observe that the slip length b is dependent on the spacing L between posts; we intu- itively expect these results to extend naturally to the case of rough surfaces, with b depending on the lateral spacing between peaks, or equivalently, the autocorrelation length A of the texture. Unlike regular arrays of posts and ridges for which the wetted solid fraction # is essentially fixed by the geom- etry of the texture, the value of # for random textures is a function also of the externally imposed liquid pressure and the loading history, and has to be ascertained through computational 164] or experimental methods [60]. Due to the lack of an accurate estimate of the Wenzel roughness rw that includes the small wavelength roughness components, # could also not be calculated from apparent contact angle measurements using equation (1.5); in any case, the value of q is expected to differ for a sessile drop in ambient air, and a submerged surface under turbulent flow. Since no reliable data measurement of # was available, the dependence of b on # could not be verified. Examining Figure 4.1, no correlation is evident between the slip length b and either the surface slope s or the Wenzel roughness rw; surfaces with comparable values of surface slope and Wenzel roughness are seen to have widely different values of slip length. We conclude, within the confidence bounds imposed by the limited data at hand, that the slip length on a rough superhydrophobic texture is not strongly dependent on either param- eter. A large Wenzel roughness, however, is critical to ensure the stability of the air-water interface on the surface, without which turbulent pressure fluctuations will quickly trigger a transition to the hydrodynamically rough

108 Wenzel state, resulting in drag increase 159,64]. Bottiglione and Carbone (2013) performed numerical simulations of wetting and interface penetration on one-dimensional, randomly generated, self-affine fractal profiles; their re- sults show that the mean square surface slope s2 is a key determinant of the breakthrough pressure pw, the externally imposed excess pressure of the liquid phase at which a catastrophic wetting transition is triggered [64]. For 2D isotropic Gaussian surfaces, they also showed that the root mean square slope and the Wenzel roughness are interrelated: V1'1 rw = 1+ exp erfc (4.10) 2 2 For large values of s, the expression on the right hand side is well approxi- mated by rw ~ (4.11) 2 Thus, a sufficiently large Wenzel roughness, or equivalently a large root mean square slope, is required to ensure that a transition to the Wenzel state does not occur even under the intense pressure fluctuations encountered in turbu- lent flow. A large Wenzel roughness also restricts the extent of penetration of the interface into the texture, so that roughness effects and the associated drag increase are minimized. Using a simple theoretical analysis, they pro- posed a design rule for robust superhydrophobicity under externally imposed pressure. Assuming the liquid-air interface to be largely flat, the interfacial (Helmholtz) free energy per unit area of the surface is simply given by

A = 0LA (1 - 0)+ USLr USAr (1 -- (4.12) where ro and r' are, respectively, the Wenzel roughness calculated for the wet and non-wet regions of the solid surface. For a Gaussian surface, the heights h(x, y) and the gradients hx(x, y) and hy(x, y) are independent uncorrelated variables, and therefore r, = r' = rw, the Wenzel roughness of the whole surface. Using the Young relation (1.2), the expression for A simplifies to

A =ULA [1 - (1 + rw Cos Oy) + wuSA (4.13) For a robust interface, the free energy must increase as the interface pene- trates deeper into the texture, increasing the wetted solid fraction. In other words, we require, dA dA > 0 (4.14) dO

109 Noticing that cos Oy < 0 for a hydrophobic substrate, the above equation yields a lower bound for the Wenzel roughness rw of the texture, 1 rw > - 1 (4.15) Cos Oy and, by extension, a corresponding lower bound on the root mean square surface slope s [64]. A sufficiently large value of Wenzel roughness, which essentially translates to a large value of the mean square slope, therefore appears necessary to stabilize the liquid-air interface against failure, particularly under the harsh conditions imposed by high Reynolds number turbulent flow. As discussed before, we also require a large autocorrelation length to obtain large slip lengths and appreciable reductions in frictional drag. However, a large value of A, in conjunction with a large value of s, necessarily entails a large value of the interface width w which conflicts with the requirement that the surface should possess low roughness, lest it should generate hydrodynamic drag. This can be seen from the fact that for a Gaussian surface, the root mean square slope is related to the second moment of the PSD function by s = IM 192,931; equation (4.5) then gives 1 w = s A (4.16)

Although equation (4.5) was derived for the particular case of H = 1, the behavior is qualitatively similar for other values of the roughness exponent; most real surfaces generated by random texturing processes typically have 0.7 < H < 1 [104]. Thus for self-affine textures, the requirements for in- terface stability (large s) and low hydrodynamic roughness (small w) seem contradictory; one can be improved only at the expense of the other. How- ever, this inference rests on the critical assumption that the surface is self- affine, with a uniform scaling behavior from the smallest to the largest length scales; a possible workaround is thus the incorporation of hierarchical rough- ness at two or more successively finer length scales into the texture. For a self-affine profile, the slope of the individual roughness components scales as k(1-2H), where k is the angular wavenumber [64,90]. Roughness gen- erated using typical texturing processes generally have a Hurst exponent of H > 0.5; thus the slope of the different spectral components of roughness decreases with decreasing wavelength. Therefore, the solid-liquid contact

110 on a random superhydrophobic texture is a combination of localized Cassie and Wenzel regimes, with the liquid bridging over the large-scale primary roughness asperities, but fully wetting the small-scale secondary roughness atop those asperities. If we now ignore the fully-wetted secondary features, the 'effective' contact angle 0, at the contact line as it traverses the 'con- tour' of the primary roughness profile can be approximated by the Wenzel equation (1.3): cos 0, = r'cosOy (4.17) Here, rw is the Wenzel roughness of the secondary texture superimposed on the primary profile. Equation (4.15) now becomes less restrictive on the Wenzel roughness of the primary profile rw: 1 1 rw > - = - (4.18) Cos 0e rw Cos Oy

The primary roughness profile can thus have a smaller mean square slope, and consequently a smaller mean square roughness, for a given lateral spac- ing between surface features. The slip length b is governed by the spacing of peaks on the primary profile, since it depends only on the extent of the free interface available between successive contact spots at which the flow neces- sarily has to come to rest; the addition of a secondary roughness is expected to have minimal influence on the slip length or drag reduction performance of the overall surface. Hierarchical texturing thus permits superhydrophobic surfaces to be designed with small roughness and large asperity spacings, without sacrificing the stability of the liquid-air interface under turbulent flow conditions. In conclusion, we finally arrive at the following general guidelines for designing scalable, randomly rough superhydrophobic surfaces for turbulent drag reduction applications: " The surface should have a sufficiently low root mean square roughness to mitigate roughness-induced hydrodynamic drag. Ideally, the scaled roughness must be below atleast an order of magnitude smaller than the thickness of the viscous sublayer in the range of operating Reynolds numbers expected in service: w+ < 0.5. " The surface texturing process must be tuned to generate large spac- ing (A >> 6,) between primary roughness features, to maximize the effective slip length and percentage friction reduction.

111 * The surface should ideally incorporate secondary (and possibly higher order) roughness features at successively smaller length scales, to ensure stability of the interface against turbulent fluctuations.

4.2 Summary and outlook

The primary objective of this thesis was to characterize scalable, randomly textured superhydrophobic surfaces for frictional drag reduction in wall- bounded turbulent flows. Accordingly, the skin friction characteristics of five prototype drag-reducing surfaces were measured in fully turbulent flow inside a custom-built Taylor-Couette apparatus, for 1.64 x 104 < Re < 8.59 x 104, and at wall shear stresses T up to 50 Pa. Frictional drag reductions of up to 26 % were obtained with randomly textured superhydrophobic surfaces, with one of the hierarchically textured variants withstanding up to 27 Pa of wall shear stress before failure. Consistent with the results of previous studies, the percentage drag reduction was found to increase with the flow Reynolds number until the onset of surface failure [54,60]. Low surface roughness, large lateral spacing between roughness peaks, and the presence of hierarchi- cal roughness features, were identified as three key design requirements for the successful deployment of scalable rough superhydrophobic surfaces for passive friction reduction in turbulent flows. Unlike the case of regularly patterned surfaces, there have been very few systematic investigations of the relationship between surface geometry and drag reduction performance of rough superhydrophobic textures in turbulent flow. Such studies are necessarily challenging because of the random na- ture of the surface texture, the intricate behavior of the liquid-air interface on the surface, the large number of variables - including external pressure, turbulence intensity, flow configuration, dissolved air concentration, process variability inherent in manufacturing random textures - all of which can significantly affect experimental results, and finally, the complexities of tur- bulence itself. These difficulties have to be surmounted before progress can be made towards a definitive understanding of the interaction between mor- phological features of rough superhydrophobic surfaces, and drag reduction capability in turbulent flow. An important factor not addressed in the present study is the longevity of the plastron on the submerged superhydrophobic texture. Evidently, this is of primary concern in real-life applications where sustained drag reduc-

112 tion performance is often expected over long durations. Even the best de- signed superhydrophobic surface will fail if the air trapped in the texture is stripped away by the flow; the depletion of the air layer can occur either by shear-induced drainage [59], or by convective mass transfer in the turbulent boundary layer [105]. Several strategies have been explored in the litera- ture to sustain drag reduction by active replenishment of the gas layer, such as supplying pressurized gas to the texture through a porous backing plate [84], vapor generation by Leidenfrost heating [106-1081, and electrochemical [109,110] or catalyst-mediated gas generation [1111. The feasibility and effi- cacy of these methods need to be investigated in high Reynolds number flows similar to those typically encountered in real-life service. Finally, an interesting question that merits further investigation is the synergy between superhydrophobic drag reduction, and two other well-known strategies for skin-friction reduction in turbulent flows, namely, streamwise riblets [6] and polymer injection [4]. Each of these drag reduction mecha- nisms operate through complex modification of turbulence flow structures, and further experimentation is required to determine if their effects can be additive under specific flow conditions. Overall, rough superhydrophobic surfaces show promise as a practical means of achieving passive drag reduction in turbulent flows; however, addi- tional progress is still required on issues of scalability, robustness and texture optimization before their full potential can be realized in real-life applica- tions.

113 114 Appendix A

Engineering drawings for the TCL device

115 00.70

I

+0.20 rnI I E I I 014 0 mm I I E I I C> 00 I I I I L,~1. Lfl 00 1000 M4 x 0.7 T 15 mm 39 14

r ------I

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lo

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CN 02.8

-I ISOMETRIC SECTIONAL VIEW ALONG DIAMETRAL SECTION 03.00

ALL DIMENSIONS ARE IN INCHES EXCEPT WHERE SPECIFIED AS mm ALUMINUM ROTOR MATERIAL AL 6061-T6 2 X 00.125 DRILL THROUGH Lu TO HOLE BELOW

C) .7

/ 0.125 LI, 8 HOLES 3-48 UNC T0.5 PITCH CIRCLE $4.75

C>

SECTION A-A Lfl Lu 1/8 NPT FEMALE 04.00

RIGHT SIDE VIEW AT FRONT VIEW 1 LOT R 1 5.25

BOTTOM VIEW RIGHT FRONT MATERIAL: ALUMINUM (6061-T6 OR OTHER SUITABLE) .0 RECESS 04.175 DEPTH 0.255 DIMENSIONS IN INCHES . In ALL

ALL CHAMFERS 0.05

BASE PLATE

mmmm 1.p /

/

/

0.125 TOP VIEW $0.1040 (#37) THRU

MATERIAL: N OPTICALLY CLEAR CAST ACRYLIC.

GLUED JOINTS TO BE TRANSPARENT o I Lfl AND WATERTIGHT UNDER NORMAL TO MODERATE PRESSURES. 0i 0i SIDES OF INNER CYLINDER AND o4 OUTER BOX TO BE OPTICALLY CLEAR N AND SCRATCH FREE.

s 8 HOLES 0 - r. [email protected] 5 00 K 5.25 04.50 A LJn 03.75 Cn CUTAWAY SECTION A-A I- -1 h, n-=;------r ------F I L ------I - 7-,o,7T I/, / I 03.10 ow Lr 0. 12 5 / ry, / ------i1--- --r ALL DIMENSIONS IN INCHES I40iF 5.25 A g 04.00 ALL CHAMFERS 0.05

SIDE VIEW L------r------r I -Tr SECTION A-A STATOR

/

/

/ I, -I \\ \ co m

------wX X > mzO nM 120 Appendix B

Experimental data

121 Step Rotor speed Rotor torque Re Re* Shear stress Q (rad s-') T s.d. (pN m) T (Pa) Smooth aluminum rotor in deionized water 1 30.20 1978.2 4.3186 16370 760.16 2.8463 2 36.31 2667.6 4.2430 19682 882.73 3.8383 3 43.65 3607.4 7.3566 23660 1026.5 5.1905 4 52.48 4889.1 6.5651 28446 1195.0 7.0347 5 63.10 6638.0 6.6636 34203 1392.5 9.5511 6 75.86 9031.7 9.1162 41119 1624.3 12.995 7 91.20 12358 34.336 49434 1900.0 17.783 8 109.6 16982 44.569 59408 2227.2 24.435 9 131.8 23578 35.999 71441 2624.4 33.925 10 158.5 32742 38.076 85914 3092.6 47.111 Smooth aluminum rotor in synthetic seawater 1 30.20 2075.6 4.7432 15597 732.20 2.9865 2 36.31 2795.6 5.9037 18753 849.76 4.0224 3 43.65 3771.4 4.0885 22544 986.98 5.4265 4 52.48 5100.0 5.3658 27104 1147.7 7.3381 5 63.10 6925.8 9.0329 32589 1337.5 9.9652 6 75.86 9431.4 9.7448 39179 1560.8 13.570 7 91.20 12877 43.387 47102 1823.8 18.528 8 109.6 17673 15.064 56604 2136.6 25.429 9 131.8 24524 19.045 68070 2516.8 35.286 Acrylic FPOSS spray-coated rotor in deionized water 1 30.20 1863.2 t 12.587 16370 737.73 2.6809 2 36.31 2503.9 7.8342 19682 855.22 3.6027 3 43.65 3344.9 11.026 23660 988.46 4.8128 4 52.48 4515.0 12.253 28446 1148.4 6.4964 5 63.10 6115.0 14.78 34203 1336.5 8.7986 6 75.86 8365.1 22.981 41119 1563.2 12.036 7 91.20 11495 49.170 49434 1832.4 16.540 Etched aluminum rotor in deionized water 1 30.20 1713.1 8.0122 16370 707.39 2.4649 2 36.31 2303.3 7.3679 19682 820.25 3.3141 3 43.65 3090.8 15.633 23660 950.18 4.4472 4 52.48 4132.4 26.389 28446 1098.7 5.9459

122 Step Rotor speed Rotor torque Re Re* Shear stress Q (rad s- 1) T I s.d. (pN m) T (Pa) Etched aluminum rotor in deionized water 5 63.10 5476.7 26.300 34203 1264.8 7.8801 6 75.86 7212.7 31.580 41119 1451.5 10.378 7 91.20 9597.5 67.591 49434 1674.4 13.809 8 109.6 12556 91.577 59408 1915.1 18.066 9 131.8 17943 107.39 71441 2289.4 25.817 10 158.5 26081 149.88 85914 2760.1 37.527 Etched aluminum rotor in synthetic seawater 1 30.20 1821.8 t 7.5161 15597 685.97 2.6213 2 36.31 2451.0 8.8668 18753 795.66 3.5266 3 43.65 3308.9 9.8911 22544 924.48 4.7610 4 52.48 4470.8 17.562 27104 1074.6 6.4328 5 63.10 6043.0 18.765 32589 1249.4 8.6950 6 75.86 8128.5 26.138 39179 1449.0 11.696 7 91.20 10833 60.557 47102 1672.8 15.587 8 109.6 14821 102.72 56604 1956.6 21.325 9 131.8 21109 87.239 68070 2335.0 30.373 10 158.5 29895 121.07 81860 2778.8 43.014 80 grit SEB rotor in deionized water 1 30.20 1856.1 5.6725 16370 736.33 2.6706 2 36.31 2446.4 7.3488 19682 845.34 3.5200 3 43.65 3251.4 8.6287 23660 974.55 4.6783 4 52.48 4288.5 11.968 28446 1119.2 6.1705 5 63.10 5664.4 13.310 34203 1286.3 8.1502 6 75.86 7471.2 19.779 41119 1477.3 10.750 7 91.20 9861.0 46.141 49434 1697.2 14.189 8 109.6 13371 49.247 59408 1976.3 19.239 9 131.8 18683 107.72 71441 2336.1 26.882 10 158.5 27170 174.87 85914 2817.2 39.094 150 grit SEB rotor in deionized water 1 30.20 1981.4 6.4166 16370 760.77 2.8509 2 36.31 2634.8 8.3094 19682 877.29 3.7911 3 43.65 3493.7 10.483 23660 1010.2 5.0269 4 52.48 4658.0 11.240 28446 1166.5 6.7022

123 Step Rotor speed Rotor torque Re Re* Shear stress Q (rad s-1) T s.d. (pN m) T (Pa) 150 grit SEB rotor in deionized water 5 63.10 6076.3 20.902 34203 1332.3 8.7429 6 75.86 8390.7 46.015 41119 1565.6 12.073 7 91.20 10180 t 79.637 49434 1724.4 14.648 8 109.6 13050 130.00 59408 1952.4 18.777 9 131.8 19158 336.33 71441 2365.6 27.566 10 158.5 29465 t 338.36 85914 2933.7 42.396 Hydrophilic grooved rotor in deionized water 1 30.20 1979.0 t 3.4682 16370 760.31 2.8475 2 36.31 2675.2 4.4119 19682 883.99 3.8492 3 43.65 3628.8 5.3581 23660 1029.6 5.2213 4 52.48 5000.4 9.3224 28446 1208.6 7.1948 5 63.10 7032.6 15.178 34203 1433.3 10.119 6 75.86 9952.7 17.113 41119 1705.1 14.320 7 91.20 14089 49.511 49434 2028.7 20.272 8 109.6 19994 40.170 59408 2416.7 28.768 9 131.8 28276 46.801 71441 2873.9 40.685 Heptane-infused grooved rotor in deionized water 1 30.20 1885.7 4.208 16370 742.17 2.7132 2 36.31 2518.1 4.7536 19682 857.64 3.6232 3 43.65 3374.8 4.8362 23660 992.87 4.8558 4 52.48 4518.6 6.2558 28446 1148.9 6.5016 5 63.10 6115.9 9.3458 34203 1336.6 8.7999 6 75.86 8333.7 29.786 41119 1560.2 11.991 7 91.20 11495 45.831 49434 1832.4 16.540 8 109.6 16126 45.507 59408 2170.4 23.203 9 131.8 23616 108.9 71441 2626.5 33.980 Air-filled grooved rotor in deionized water 1 30.20 1857.4 4.1954 16370 736.58 2.6725 2 36.31 2453.3 3.9672 19682 846.53 3.5299 3 43.65 3254.2 7.0407 23660 974.97 4.6823 4 52.48 4318.5 8.6891 28446 1123.1 6.2137 5 63.10 5809.9 6.3313 34203 1302.7 8.3596 6 75.86 7924.4 10.035 41119 1521.4 11.402

124 Step Rotor speed Rotor torque Re Re* Shear stress Q (rad s-') T s.d. (pN m) T (Pa) Air-filled grooved rotor in deionized water 7 91.20 10832 34.096 49434 1778.8 15.586 8 109.6 15028 17.618 59408 2095.2 21.623 9 131.8 21215 24.613 71441 2489.4 30.525 10 158.5 31252 195.72 85914 3021.4 44.967 Air-filled grooved rotor in synthetic seawater 1 30.20 1917.7 2.6056 15597 703.80 2.7593 2 36.31 2565.1 2.9507 18753 813.97 3.6908 3 43.65 3441.6 3.1543 22544 942.84 4.9519 4 52.48 4622.8 4.3985 27104 1092.7 6.6515 5 63.10 6244.9 6.8926 32589 1270.1 8.9855 6 75.86 8468.9 10.593 39179 1479.0 12.186 7 91.20 11539 33.093 47102 1726.4 16.603 8 109.6 15812 16.466 56604 2020.9 22.751 9 131.8 22073 20.311 68070 2387.8 31.760

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